حمّل تطبيق منهاجي الجديد

منهاجي صار أسرع من خلال التطبيق

  أتدرب وأحل المسائل

أتدرب وأحل المسائل

التكامل بالكسور الجزئية

أجد كلاً من التكاملات الآتية:

begin mathsize 20px style integral fraction numerator x minus 10 over denominator x left parenthesis x plus 5 right parenthesis end fraction d x end style (1)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator x minus 10 over denominator x left parenthesis x plus 5 right parenthesis end fraction equals A over x plus fraction numerator B over denominator x plus 5 end fraction end cell row blank cell not stretchy rightwards double arrow x minus 10 equals A left parenthesis x plus 5 right parenthesis plus B x end cell row blank cell x equals 0 not stretchy rightwards double arrow A equals negative 2 end cell row blank cell table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell x equals negative 5 not stretchy rightwards double arrow B equals 3 end cell end table end cell row blank cell integral fraction numerator x minus 10 over denominator x left parenthesis x plus 5 right parenthesis end fraction d x equals integral left parenthesis fraction numerator negative 2 over denominator x end fraction plus fraction numerator 3 over denominator x plus 5 end fraction right parenthesis d x end cell row blank cell space of 1em equals negative 2 ln invisible function application vertical line x vertical line plus 3 ln invisible function application vertical line x plus 5 vertical line plus C end cell end table end style

begin mathsize 20px style integral fraction numerator 2 over denominator 1 minus x squared end fraction d x end style (2)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 2 over denominator 1 minus x squared end fraction equals fraction numerator 2 over denominator left parenthesis 1 minus x right parenthesis left parenthesis 1 plus x right parenthesis end fraction equals fraction numerator A over denominator 1 minus x end fraction plus fraction numerator B over denominator 1 plus x end fraction end cell row blank cell not stretchy rightwards double arrow 2 equals A left parenthesis 1 plus x right parenthesis plus B left parenthesis 1 minus x right parenthesis end cell row blank cell x equals 1 not stretchy rightwards double arrow A equals 1 end cell row blank cell table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell x equals negative 1 not stretchy rightwards double arrow B end cell cell equals 1 end cell row cell integral fraction numerator 2 over denominator 1 minus x squared end fraction d x end cell cell equals integral left parenthesis fraction numerator 1 over denominator 1 minus x end fraction plus fraction numerator 1 over denominator 1 plus x end fraction right parenthesis d x end cell row blank cell equals negative ln invisible function application vertical line 1 minus x vertical line plus ln invisible function application vertical line 1 plus x vertical line plus C end cell row blank cell equals ln invisible function application vertical line fraction numerator 1 plus x over denominator 1 minus x end fraction vertical line plus C end cell end table end cell end table end style

begin mathsize 20px style integral fraction numerator 4 over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 4 right parenthesis end fraction d x end style (3)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 4 over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 4 right parenthesis end fraction equals fraction numerator A over denominator x minus 2 end fraction plus fraction numerator B over denominator x minus 4 end fraction end cell row blank cell not stretchy rightwards double arrow 4 equals A left parenthesis x minus 4 right parenthesis plus B left parenthesis x minus 2 right parenthesis end cell row blank cell x equals 2 not stretchy rightwards double arrow A equals negative 2 end cell row blank cell x equals 4 not stretchy rightwards double arrow B equals 2 end cell row blank cell integral fraction numerator 4 over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 4 right parenthesis end fraction d x equals integral left parenthesis fraction numerator negative 2 over denominator x minus 2 end fraction plus fraction numerator 2 over denominator x minus 4 end fraction right parenthesis d x end cell row blank cell equals negative 2 ln invisible function application vertical line x minus 2 vertical line plus 2 ln invisible function application vertical line x minus 4 vertical line plus C end cell row blank cell equals 2 ln invisible function application vertical line fraction numerator x minus 4 over denominator x minus 2 end fraction vertical line plus C end cell end table end style

begin mathsize 20px style integral fraction numerator 3 x plus 4 over denominator x squared plus x end fraction d x end style (4)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 3 x plus 4 over denominator x squared plus x end fraction equals fraction numerator 3 x plus 4 over denominator x left parenthesis x plus 1 right parenthesis end fraction equals A over x plus fraction numerator B over denominator x plus 1 end fraction end cell row blank cell not stretchy rightwards double arrow 3 x plus 4 equals A left parenthesis x plus 1 right parenthesis plus B x end cell row blank cell x equals 0 not stretchy rightwards double arrow A equals 4 end cell row blank cell x equals negative 1 not stretchy rightwards double arrow B equals negative 1 end cell row blank cell table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell fraction numerator 3 x plus 4 over denominator x squared plus x end fraction d x end cell cell equals integral left parenthesis 4 over x plus fraction numerator negative 1 over denominator x plus 1 end fraction right parenthesis d x end cell row blank cell equals 4 ln invisible function application vertical line x vertical line minus ln invisible function application vertical line x plus 1 vertical line plus C end cell end table end cell end table end style

begin mathsize 20px style integral fraction numerator x squared over denominator x squared minus 4 end fraction d x end style (5)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell integral fraction numerator x squared over denominator x squared minus 4 end fraction d x equals integral left parenthesis 1 plus fraction numerator 4 over denominator x squared minus 4 end fraction right parenthesis d x end cell row blank cell fraction numerator 4 over denominator x squared minus 4 end fraction equals fraction numerator 4 over denominator left parenthesis x minus 2 right parenthesis left parenthesis x plus 2 right parenthesis end fraction equals fraction numerator A over denominator x minus 2 end fraction plus fraction numerator B over denominator x plus 2 end fraction end cell row blank cell not stretchy rightwards double arrow 4 equals A left parenthesis x plus 2 right parenthesis plus B left parenthesis x minus 2 right parenthesis end cell row blank cell table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell x equals 2 not stretchy rightwards double arrow A end cell cell equals 1 end cell row cell x equals negative 2 not stretchy rightwards double arrow B end cell cell equals negative 1 end cell row cell integral fraction numerator x squared over denominator x squared minus 4 end fraction d x end cell cell equals integral left parenthesis 1 plus fraction numerator 1 over denominator x minus 2 end fraction plus fraction numerator negative 1 over denominator x plus 2 end fraction right parenthesis d x end cell row blank cell equals x plus l n invisible function application vertical line x minus 2 vertical line minus l n invisible function application vertical line x plus 2 vertical line plus C end cell row blank cell equals x plus l n invisible function application vertical line fraction numerator x minus 2 over denominator x plus 2 end fraction vertical line plus C end cell end table end cell end table end style

begin mathsize 20px style integral fraction numerator 3 x minus 6 over denominator x squared plus x minus 2 end fraction d x end style (6)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 3 x minus 6 over denominator x squared plus x minus 2 end fraction equals fraction numerator 3 x minus 6 over denominator left parenthesis x plus 2 right parenthesis left parenthesis x minus 1 right parenthesis end fraction equals fraction numerator A over denominator x plus 2 end fraction plus fraction numerator B over denominator x minus 1 end fraction end cell row blank cell not stretchy rightwards double arrow 3 x minus 6 equals A left parenthesis x minus 1 right parenthesis plus B left parenthesis x plus 2 right parenthesis end cell row blank cell x equals negative 2 not stretchy rightwards double arrow A equals 4 end cell row blank cell x equals 1 not stretchy rightwards double arrow B equals negative 1 end cell row blank cell table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell x fraction numerator 3 x minus 6 over denominator x squared plus x minus 2 end fraction d x equals integral left parenthesis fraction numerator 4 over denominator x plus 2 end fraction plus fraction numerator negative 1 over denominator x minus 1 end fraction right parenthesis d x end cell row blank cell equals 4 ln invisible function application vertical line x plus 2 vertical line minus ln invisible function application vertical line x minus 1 vertical line plus C end cell end table end cell end table end style

begin mathsize 20px style integral fraction numerator 4 x plus 10 over denominator 4 x squared minus 4 x minus 3 end fraction d x end style (7)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 4 x plus 10 over denominator 4 x squared minus 4 x minus 3 end fraction equals fraction numerator 4 x plus 10 over denominator left parenthesis 2 x minus 3 right parenthesis left parenthesis 2 x plus 1 right parenthesis end fraction equals fraction numerator A over denominator 2 x minus 3 end fraction plus fraction numerator B over denominator 2 x plus 1 end fraction end cell row blank cell not stretchy rightwards double arrow 4 x plus 10 equals A left parenthesis 2 x plus 1 right parenthesis plus B left parenthesis 2 x minus 3 right parenthesis end cell row blank cell x equals 3 over 2 not stretchy rightwards double arrow A equals 4 end cell row blank cell x equals negative 1 half not stretchy rightwards double arrow B equals negative 2 end cell row blank cell integral fraction numerator 4 x plus 10 over denominator 4 x squared minus 4 x minus 3 end fraction d x equals integral left parenthesis fraction numerator 4 over denominator 2 x minus 3 end fraction plus fraction numerator negative 2 over denominator 2 x plus 1 end fraction right parenthesis d x end cell row blank cell equals 2 ln invisible function application vertical line 2 x minus 3 vertical line minus ln invisible function application vertical line 2 x plus 1 vertical line plus C end cell end table end style

begin mathsize 20px style integral fraction numerator 2 x squared plus 9 x minus 11 over denominator x cubed plus 2 x squared minus 5 x minus 6 end fraction d x end style (8)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 2 x squared plus 9 x minus 11 over denominator x cubed plus 2 x squared minus 5 x minus 6 end fraction equals fraction numerator 2 x squared plus 9 x minus 11 over denominator left parenthesis x minus 2 right parenthesis left parenthesis x plus 1 right parenthesis left parenthesis x plus 3 right parenthesis end fraction equals fraction numerator A over denominator x minus 2 end fraction plus fraction numerator B over denominator x plus 1 end fraction plus fraction numerator C over denominator x plus 3 end fraction end cell row blank cell not stretchy rightwards double arrow 2 x squared plus 9 x minus 11 equals A left parenthesis x plus 1 right parenthesis left parenthesis x plus 3 right parenthesis plus B left parenthesis x minus 2 right parenthesis left parenthesis x plus 3 right parenthesis plus C left parenthesis x minus 2 right parenthesis left parenthesis x plus 1 right parenthesis end cell row blank cell x equals 2 not stretchy rightwards double arrow A equals 1 end cell row blank cell x equals negative 1 not stretchy rightwards double arrow B equals 3 end cell row blank cell x equals negative 3 not stretchy rightwards double arrow C equals negative 2 end cell row blank cell integral fraction numerator 2 x squared plus 9 x minus 11 over denominator x cubed plus 2 x squared minus 5 x minus 6 end fraction d x equals integral left parenthesis fraction numerator 1 over denominator x minus 2 end fraction plus fraction numerator 3 over denominator x plus 1 end fraction plus fraction numerator negative 2 over denominator x plus 3 end fraction right parenthesis d x end cell row blank cell equals ln invisible function application vertical line x minus 2 vertical line plus 3 ln invisible function application vertical line x plus 1 vertical line minus 2 ln invisible function application vertical line x plus 3 vertical line plus C end cell end table end style

begin mathsize 20px style integral fraction numerator 4 x over denominator x squared minus 2 x minus 3 end fraction d x end style (9)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 4 x over denominator x squared minus 2 x minus 3 end fraction equals fraction numerator 4 x over denominator left parenthesis x minus 3 right parenthesis left parenthesis x plus 1 right parenthesis end fraction equals fraction numerator A over denominator x minus 3 end fraction plus fraction numerator B over denominator x plus 1 end fraction end cell row blank cell not stretchy rightwards double arrow 4 x equals A left parenthesis x plus 1 right parenthesis plus B left parenthesis x minus 3 right parenthesis end cell row blank cell x equals 3 not stretchy rightwards double arrow A equals 3 end cell row blank cell x equals negative 1 not stretchy rightwards double arrow B equals 1 end cell row blank cell integral fraction numerator 4 x over denominator x squared minus 2 x minus 3 end fraction d x equals integral left parenthesis fraction numerator 3 over denominator x minus 3 end fraction plus fraction numerator 1 over denominator x plus 1 end fraction right parenthesis d x end cell row blank cell equals 3 ln invisible function application vertical line x minus 3 vertical line plus ln invisible function application vertical line x plus 1 vertical line plus C end cell end table end style

begin mathsize 20px style integral fraction numerator 8 x squared minus 19 x plus 1 over denominator left parenthesis 2 x plus 1 right parenthesis left parenthesis x minus 2 right parenthesis squared end fraction d x end style (10)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 8 x squared minus 19 x plus 1 over denominator left parenthesis 2 x plus 1 right parenthesis left parenthesis x minus 2 right parenthesis squared end fraction equals fraction numerator A over denominator 2 x plus 1 end fraction plus fraction numerator B over denominator x minus 2 end fraction plus fraction numerator C over denominator left parenthesis x minus 2 right parenthesis squared end fraction end cell row blank cell not stretchy rightwards double arrow 8 x squared minus 19 x plus 1 equals A left parenthesis x minus 2 right parenthesis squared plus B left parenthesis 2 x plus 1 right parenthesis left parenthesis x minus 2 right parenthesis plus C left parenthesis 2 x plus 1 right parenthesis end cell row blank cell x equals negative 1 half not stretchy rightwards double arrow A equals 2 end cell row blank cell x equals 2 not stretchy rightwards double arrow C equals negative 1 end cell row blank cell x equals 0 not stretchy rightwards double arrow 1 equals 4 A minus 2 B plus C not stretchy rightwards double arrow B equals 3 end cell row blank cell integral fraction numerator 8 x squared minus 19 x plus 1 over denominator left parenthesis 2 x plus 1 right parenthesis left parenthesis x minus 2 right parenthesis squared end fraction d x equals integral left parenthesis fraction numerator 2 over denominator 2 x plus 1 end fraction plus fraction numerator 3 over denominator x minus 2 end fraction plus fraction numerator negative 1 over denominator left parenthesis x minus 2 right parenthesis squared end fraction right parenthesis d x end cell row blank cell space of 1em equals ln invisible function application vertical line 2 x plus 1 vertical line plus 3 ln invisible function application vertical line x minus 2 vertical line plus fraction numerator 1 over denominator x minus 2 end fraction plus C end cell end table end style

begin mathsize 20px style integral fraction numerator 9 x squared minus 3 x plus 2 over denominator 9 x squared minus 4 end fraction d x end style (11)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell integral fraction numerator 9 x squared minus 3 x plus 2 over denominator 9 x squared minus 4 end fraction d x equals integral left parenthesis 1 plus fraction numerator 6 minus 3 x over denominator 9 x squared minus 4 end fraction right parenthesis d x end cell row blank cell fraction numerator 6 minus 3 x over denominator 9 x squared minus 4 end fraction equals fraction numerator 6 minus 3 x over denominator left parenthesis 3 x minus 2 right parenthesis left parenthesis 3 x plus 2 right parenthesis end fraction equals fraction numerator A over denominator 3 x minus 2 end fraction plus fraction numerator B over denominator 3 x plus 2 end fraction end cell row blank cell not stretchy rightwards double arrow 6 minus 3 x equals A left parenthesis 3 x plus 2 right parenthesis plus B left parenthesis 3 x minus 2 right parenthesis end cell row blank cell x equals 2 over 3 not stretchy rightwards double arrow A equals 1 end cell row blank cell x equals negative 2 over 3 not stretchy rightwards double arrow B equals negative 2 end cell row blank cell integral fraction numerator 9 x squared minus 3 x plus 2 over denominator 9 x squared minus 4 end fraction d x equals integral left parenthesis 1 plus fraction numerator 1 over denominator 3 x minus 2 end fraction plus fraction numerator negative 2 over denominator 3 x plus 2 end fraction right parenthesis d x end cell row blank cell equals x plus 1 third ln invisible function application vertical line 3 x minus 2 vertical line minus 2 over 3 ln invisible function application vertical line 3 x plus 2 vertical line plus C end cell end table end style

begin mathsize 20px style integral fraction numerator x cubed plus 2 x squared plus 2 over denominator x squared plus x end fraction d x end style (12)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell integral fraction numerator x cubed plus 2 x squared plus 2 over denominator x squared plus x end fraction d x equals integral left parenthesis x plus 1 plus fraction numerator 2 minus x over denominator x squared plus x end fraction right parenthesis d x end cell row blank cell fraction numerator 2 minus x over denominator x squared plus x end fraction equals fraction numerator 2 minus x over denominator x left parenthesis x plus 1 right parenthesis end fraction equals A over x plus fraction numerator B over denominator x plus 1 end fraction end cell row blank cell not stretchy rightwards double arrow 2 minus x equals A left parenthesis x plus 1 right parenthesis plus B x end cell row blank cell x equals 0 not stretchy rightwards double arrow A equals 2 end cell row blank cell x equals negative 1 not stretchy rightwards double arrow B equals negative 3 end cell row blank cell integral fraction numerator x cubed plus 2 x squared plus 2 over denominator x squared plus x end fraction d x equals integral left parenthesis x plus 1 plus 2 over x plus fraction numerator negative 3 over denominator x plus 1 end fraction right parenthesis d x end cell row blank cell equals 1 half x squared plus x plus 2 ln invisible function application vertical line x vertical line minus 3 ln invisible function application vertical line x plus 1 vertical line plus C end cell end table end style

begin mathsize 20px style integral fraction numerator x squared plus x plus 2 over denominator 3 minus 2 x minus x squared end fraction d x end style (13)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell integral fraction numerator x squared plus x plus 2 over denominator 3 minus 2 x minus x squared end fraction d x equals integral left parenthesis negative 1 plus fraction numerator 5 minus x over denominator negative x squared minus 2 x plus 3 end fraction right parenthesis d x end cell row blank cell fraction numerator 5 minus x over denominator negative x squared minus 2 x plus 3 end fraction equals fraction numerator x minus 5 over denominator left parenthesis x plus 3 right parenthesis left parenthesis x minus 1 right parenthesis end fraction equals fraction numerator A over denominator x plus 3 end fraction plus fraction numerator B over denominator x minus 1 end fraction end cell row blank cell not stretchy rightwards double arrow x minus 5 equals A left parenthesis x minus 1 right parenthesis plus B left parenthesis x plus 3 right parenthesis end cell row blank cell x equals negative 3 not stretchy rightwards double arrow A equals 2 end cell row blank cell table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell x equals 1 not stretchy rightwards double arrow B equals negative 1 end cell end table end cell row blank cell integral fraction numerator x squared plus x plus 2 over denominator 3 minus 2 x minus x squared end fraction d x equals integral left parenthesis negative 1 plus fraction numerator 2 over denominator x plus 3 end fraction plus fraction numerator negative 1 over denominator x minus 1 end fraction right parenthesis d x end cell row blank cell equals negative x plus 2 ln invisible function application vertical line x plus 3 vertical line minus ln invisible function application vertical line x minus 1 vertical line plus C end cell end table end style

begin mathsize 20px style integral fraction numerator 2 x minus 4 over denominator left parenthesis x squared plus 4 right parenthesis left parenthesis x plus 2 right parenthesis end fraction d x end style (14)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 2 x minus 4 over denominator left parenthesis x squared plus 4 right parenthesis left parenthesis x plus 2 right parenthesis end fraction equals fraction numerator A over denominator x plus 2 end fraction plus fraction numerator B x plus C over denominator x squared plus 4 end fraction end cell row blank cell not stretchy rightwards double arrow 2 x minus 4 equals A left parenthesis x squared plus 4 right parenthesis plus left parenthesis B x plus C right parenthesis left parenthesis x plus 2 right parenthesis end cell row blank cell x equals negative 2 not stretchy rightwards double arrow A equals negative 1 end cell row blank cell x equals 0 not stretchy rightwards double arrow negative 4 equals 4 A plus 2 C not stretchy rightwards double arrow C equals 0 end cell row blank cell x equals 1 not stretchy rightwards double arrow negative 2 equals 5 A plus 3 B plus 3 C not stretchy rightwards double arrow B equals 1 end cell row blank cell integral fraction numerator 2 x minus 4 over denominator left parenthesis x squared plus 4 right parenthesis left parenthesis x plus 2 right parenthesis end fraction d x equals integral left parenthesis fraction numerator negative 1 over denominator x plus 2 end fraction plus fraction numerator x over denominator x squared plus 4 end fraction right parenthesis d x end cell row blank cell equals negative ln invisible function application vertical line x plus 2 vertical line plus 1 half ln invisible function application vertical line x squared plus 4 vertical line plus C end cell end table end style

begin mathsize 20px style integral fraction numerator x cubed minus 4 x squared minus 2 over denominator x cubed plus x squared end fraction d x end style (15)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell integral fraction numerator x cubed minus 4 x squared minus 2 over denominator x cubed plus x squared end fraction d x equals integral left parenthesis 1 plus fraction numerator negative 5 x squared minus 2 over denominator x cubed plus x squared end fraction right parenthesis d x end cell row blank cell fraction numerator negative 5 x squared minus 2 over denominator x cubed plus x squared end fraction equals fraction numerator negative 5 x squared minus 2 over denominator x squared left parenthesis x plus 1 right parenthesis end fraction equals A over x plus B over x squared plus fraction numerator C over denominator x plus 1 end fraction end cell row blank cell not stretchy rightwards double arrow negative 5 x squared minus 2 equals A x left parenthesis x plus 1 right parenthesis plus B left parenthesis x plus 1 right parenthesis plus C x squared end cell row blank cell x equals 0 not stretchy rightwards double arrow B equals negative 2 end cell row blank cell x equals negative 1 not stretchy rightwards double arrow C equals negative 7 end cell row blank cell x equals 1 not stretchy rightwards double arrow negative 7 equals 2 A plus 2 B plus C not stretchy rightwards double arrow A equals 2 end cell row blank cell integral fraction numerator x cubed minus 4 x squared minus 2 over denominator x cubed plus x squared end fraction d x equals integral left parenthesis 1 plus 2 over x plus fraction numerator negative 2 over denominator x squared end fraction plus fraction numerator negative 7 over denominator x plus 1 end fraction right parenthesis d x end cell row blank cell equals x plus 2 ln invisible function application vertical line x vertical line plus 2 over x minus 7 ln invisible function application vertical line x plus 1 vertical line plus C end cell end table end style

begin mathsize 20px style integral fraction numerator 3 minus x over denominator 2 minus 5 x minus 12 x squared end fraction d x end style (16)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 3 minus x over denominator 2 minus 5 x minus 12 x squared end fraction equals fraction numerator x minus 3 over denominator 12 x squared plus 5 x minus 2 end fraction equals fraction numerator x minus 3 over denominator left parenthesis 4 x minus 1 right parenthesis left parenthesis 3 x plus 2 right parenthesis end fraction equals fraction numerator A over denominator 4 x minus 1 end fraction plus fraction numerator B over denominator 3 x plus 2 end fraction end cell row blank cell not stretchy rightwards double arrow x minus 3 equals A left parenthesis 3 x plus 2 right parenthesis plus B left parenthesis 4 x minus 1 right parenthesis end cell row blank cell x equals 1 fourth not stretchy rightwards double arrow A equals negative 1 end cell row blank cell x equals negative 2 over 3 not stretchy rightwards double arrow B equals 1 end cell row blank cell integral fraction numerator 3 minus x over denominator 2 minus 5 x minus 12 x squared end fraction d x equals integral left parenthesis fraction numerator negative 1 over denominator 4 x minus 1 end fraction plus fraction numerator 1 over denominator 3 x plus 2 end fraction right parenthesis d x end cell row blank cell equals negative 1 fourth ln invisible function application vertical line 4 x minus 1 vertical line plus 1 third ln invisible function application vertical line 3 x plus 2 vertical line plus C end cell end table end style

begin mathsize 20px style integral fraction numerator 3 x cubed minus x squared plus 12 x minus 6 over denominator x to the power of 4 plus 6 x squared end fraction d x end style (17)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 3 x cubed minus x squared plus 12 x minus 6 over denominator x to the power of 4 plus 6 x squared end fraction equals fraction numerator 3 x cubed minus x squared plus 12 x minus 6 over denominator x squared left parenthesis x squared plus 6 right parenthesis end fraction equals A over x plus B over x squared plus fraction numerator C x plus D over denominator x squared plus 6 end fraction end cell row blank cell not stretchy rightwards double arrow 3 x cubed minus x squared plus 12 x minus 6 equals A x left parenthesis x squared plus 6 right parenthesis plus B left parenthesis x squared plus 6 right parenthesis plus left parenthesis C x plus D right parenthesis left parenthesis x squared right parenthesis end cell row blank cell x equals 0 not stretchy rightwards double arrow B equals negative 1 end cell row blank cell x equals 1 not stretchy rightwards double arrow 8 equals 7 A plus 7 B plus C plus D horizontal ellipsis horizontal ellipsis horizontal ellipsis horizontal ellipsis horizontal ellipsis left parenthesis 1 right parenthesis end cell row blank cell x equals negative 1 not stretchy rightwards double arrow negative 22 equals negative 7 A plus 7 B minus C plus D horizontal ellipsis horizontal ellipsis left parenthesis 2 right parenthesis end cell row blank cell x equals 2 not stretchy rightwards double arrow 38 equals 20 A plus 10 B plus 8 C plus 4 D horizontal ellipsis horizontal ellipsis text (3) end text end cell end table end style

 بجمع (1)، (2) ينتج أن: begin mathsize 20px style 14 B plus 2 D equals negative 14 end style، وبتعويض begin mathsize 20px style B equals negative 1 end style، نجد أن begin mathsize 20px style D equals 0 end style

وبطرح (2) من (1) ينتج أن begin mathsize 20px style 14 A plus 2 C equals 30 end style أي أن begin mathsize 20px style C equals 15 minus 7 A end style

بالتعويض في (3) ينتج أن: 

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell 20 A minus 10 plus 8 left parenthesis 15 minus 7 A right parenthesis end cell cell equals 38 end cell row cell negative 36 A end cell cell equals negative 72 not stretchy rightwards double arrow A equals 2 end cell row C cell equals 15 minus 7 left parenthesis 2 right parenthesis equals 1 end cell end table
table attributes columnalign right columnspacing 1em end attributes row cell integral fraction numerator 3 x cubed minus x squared plus 12 x minus 6 over denominator x to the power of 4 plus 6 x squared end fraction d x equals integral left parenthesis 2 over x plus fraction numerator negative 1 over denominator x squared end fraction plus fraction numerator x over denominator x squared plus 6 end fraction right parenthesis d x end cell row cell equals 2 l n invisible function application vertical line x vertical line plus 1 over x plus 1 half l n invisible function application vertical line x squared plus 6 vertical line plus C end cell end table end style

begin mathsize 20px style integral fraction numerator 5 x minus 2 over denominator left parenthesis x minus 2 right parenthesis squared end fraction d x end style (18)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 5 x minus 2 over denominator left parenthesis x minus 2 right parenthesis squared end fraction equals fraction numerator A over denominator x minus 2 end fraction plus fraction numerator B over denominator left parenthesis x minus 2 right parenthesis squared end fraction end cell row blank cell not stretchy rightwards double arrow 5 x minus 2 equals A left parenthesis x minus 2 right parenthesis plus B end cell row blank cell x equals 2 not stretchy rightwards double arrow B equals 8 end cell row blank cell x equals 0 not stretchy rightwards double arrow negative 2 equals negative 2 A plus B not stretchy rightwards double arrow A equals 5 end cell row blank cell integral fraction numerator 5 x minus 2 over denominator left parenthesis x minus 2 right parenthesis squared end fraction d x equals integral left parenthesis fraction numerator 5 over denominator x minus 2 end fraction plus fraction numerator 8 over denominator left parenthesis x minus 2 right parenthesis squared end fraction right parenthesis d x end cell row blank cell equals 5 ln invisible function application vertical line x minus 2 vertical line minus fraction numerator 8 over denominator x minus 2 end fraction plus C end cell end table end style

ملاحظة: يمكن حل هذا التكامل بالتعويض begin mathsize 20px style u equals x minus 2 end style

كما يمكن حله بالأجزاء حيث: begin mathsize 20px style u equals 5 x minus 2 comma d v equals left parenthesis x minus 2 right parenthesis to the power of negative 2 end exponent end style

أجد قيمة كل من التكاملات الآتية:

begin mathsize 20px style integral subscript 2 superscript 4 fraction numerator 6 plus 3 x minus x squared over denominator x cubed plus 2 x squared end fraction d x end style (19)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 6 plus 3 x minus x squared over denominator x cubed plus 2 x squared end fraction equals fraction numerator 6 plus 3 x minus x squared over denominator x squared left parenthesis x plus 2 right parenthesis end fraction equals A over x plus B over x squared plus fraction numerator C over denominator x plus 2 end fraction end cell row blank cell not stretchy rightwards double arrow 6 plus 3 x minus x squared equals A x left parenthesis x plus 2 right parenthesis plus B left parenthesis x plus 2 right parenthesis plus C left parenthesis x squared right parenthesis end cell row blank cell x equals 0 not stretchy rightwards double arrow B equals 3 end cell row blank cell x equals negative 2 not stretchy rightwards double arrow C equals negative 1 end cell row blank cell x equals 1 not stretchy rightwards double arrow 8 equals 3 A plus 3 B plus C not stretchy rightwards double arrow A equals 0 end cell row blank cell table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell integral subscript 2 superscript 4 fraction numerator 6 plus 3 x minus x squared over denominator x cubed plus 2 x squared end fraction d x end cell cell equals integral subscript 2 superscript 4 left parenthesis 3 over x squared plus fraction numerator negative 1 over denominator x plus 2 end fraction right parenthesis d x end cell row blank cell equals left parenthesis negative 3 over x minus ln invisible function application vertical line x plus 2 vertical line right parenthesis vertical line subscript 2 superscript 4 end cell row blank cell equals negative 3 over 4 minus ln invisible function application 6 plus 3 over 2 plus ln invisible function application 4 equals 3 over 4 plus ln invisible function application 2 over 3 end cell end table end cell end table end style

begin mathsize 20px style integral subscript negative 1 divided by 3 end subscript superscript 1 divided by 3 end superscript fraction numerator 9 x squared plus 4 over denominator 9 x squared minus 4 end fraction d x end style (20)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 9 x squared plus 4 over denominator 9 x squared minus 4 end fraction equals 1 plus fraction numerator 8 over denominator 9 x squared minus 4 end fraction end cell row blank cell fraction numerator 8 over denominator 9 x squared minus 4 end fraction equals fraction numerator 8 over denominator left parenthesis 3 x minus 2 right parenthesis left parenthesis 3 x plus 2 right parenthesis end fraction equals fraction numerator A over denominator 3 x minus 2 end fraction plus fraction numerator B over denominator 3 x plus 2 end fraction end cell row blank cell not stretchy rightwards double arrow 8 equals A left parenthesis 3 x plus 2 right parenthesis plus B left parenthesis 3 x minus 2 right parenthesis end cell row blank cell x equals 2 over 3 not stretchy rightwards double arrow A equals 2 end cell row blank cell x equals negative 2 over 3 not stretchy rightwards double arrow B equals negative 2 end cell row blank cell integral subscript negative 1 third end subscript superscript 1 third end superscript fraction numerator 9 x squared plus 4 over denominator 9 x squared minus 4 end fraction d x equals integral subscript negative 1 third end subscript superscript 1 third end superscript left parenthesis 1 plus fraction numerator 2 over denominator 3 x minus 2 end fraction plus fraction numerator negative 2 over denominator 3 x plus 2 end fraction right parenthesis d x end cell row blank cell equals left parenthesis x plus 2 over 3 ln invisible function application vertical line 3 x minus 2 vertical line minus 2 over 3 ln invisible function application vertical line 3 x plus 2 vertical line right parenthesis vertical line subscript negative 1 third end subscript superscript 1 third end superscript end cell row blank cell equals left parenthesis x plus 2 over 3 ln invisible function application vertical line fraction numerator 3 x minus 2 over denominator 3 x plus 2 end fraction vertical line right parenthesis vertical line subscript negative 1 third end subscript superscript 1 third end superscript end cell row blank cell equals 1 third plus 2 over 3 ln invisible function application 1 third plus 1 third minus 2 over 3 ln invisible function application 3 equals 2 over 3 minus 4 over 3 ln invisible function application 3 end cell row blank blank end table end style

begin mathsize 20px style integral subscript 0 superscript 1 fraction numerator 17 minus 5 x over denominator left parenthesis 2 x plus 3 right parenthesis left parenthesis 2 minus x right parenthesis squared end fraction d x end style (21)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 17 minus 5 x over denominator left parenthesis 2 x plus 3 right parenthesis left parenthesis 2 minus x right parenthesis squared end fraction equals fraction numerator A over denominator 2 x plus 3 end fraction plus fraction numerator B over denominator 2 minus x end fraction plus fraction numerator C over denominator left parenthesis 2 minus x right parenthesis squared end fraction end cell row blank cell not stretchy rightwards double arrow 17 minus 5 x equals A left parenthesis 2 minus x right parenthesis squared plus B left parenthesis 2 minus x right parenthesis left parenthesis 2 x plus 3 right parenthesis plus C left parenthesis 2 x plus 3 right parenthesis end cell row blank cell x equals negative 3 over 2 not stretchy rightwards double arrow A equals 2 end cell row blank cell x equals 2 not stretchy rightwards double arrow C equals 1 end cell row blank cell x equals 0 not stretchy rightwards double arrow 17 equals 4 A plus 6 B plus 3 C not stretchy rightwards double arrow B equals 1 end cell row blank cell integral subscript 0 superscript 1 fraction numerator 17 minus 5 x over denominator left parenthesis 2 x plus 3 right parenthesis left parenthesis 2 minus x right parenthesis squared end fraction d x equals integral subscript 0 superscript 1 left parenthesis fraction numerator 2 over denominator 2 x plus 3 end fraction plus fraction numerator 1 over denominator 2 minus x end fraction plus fraction numerator 1 over denominator left parenthesis 2 minus x right parenthesis squared end fraction right parenthesis d x end cell row blank cell equals left parenthesis l n invisible function application vertical line 2 x plus 3 vertical line minus l n invisible function application vertical line 2 minus x vertical line plus fraction numerator 1 over denominator 2 minus x end fraction right parenthesis vertical line subscript 0 superscript 1 end cell row blank cell equals l n invisible function application 5 plus 1 minus l n invisible function application 3 plus l n invisible function application 2 minus 1 half equals 1 half plus l n invisible function application 10 over 3 end cell end table end style

begin mathsize 20px style integral subscript 1 superscript 4 fraction numerator 4 over denominator 16 x squared plus 8 x minus 3 end fraction d x end style (22)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 4 over denominator 16 x squared plus 8 x minus 3 end fraction equals fraction numerator 4 over denominator left parenthesis 4 x minus 1 right parenthesis left parenthesis 4 x plus 3 right parenthesis end fraction equals fraction numerator A over denominator 4 x minus 1 end fraction plus fraction numerator B over denominator 4 x plus 3 end fraction end cell row blank cell not stretchy rightwards double arrow 4 equals A left parenthesis 4 x plus 3 right parenthesis plus B left parenthesis 4 x minus 1 right parenthesis end cell row blank cell x equals 1 fourth not stretchy rightwards double arrow A equals 1 end cell row blank cell x equals negative 3 over 4 not stretchy rightwards double arrow B equals negative 1 end cell row blank cell integral subscript 1 superscript 4 fraction numerator 4 over denominator 16 x squared plus 8 x minus 3 end fraction d x equals integral subscript 1 superscript 4 left parenthesis fraction numerator 1 over denominator 4 x minus 1 end fraction plus fraction numerator negative 1 over denominator 4 x plus 3 end fraction right parenthesis d x end cell row blank cell equals left parenthesis 1 fourth ln invisible function application vertical line 4 x minus 1 vertical line minus 1 fourth ln invisible function application vertical line 4 x plus 3 vertical line right parenthesis vertical line subscript 1 superscript 4 end cell row blank cell equals left parenthesis 1 fourth ln invisible function application vertical line fraction numerator 4 x minus 1 over denominator 4 x plus 3 end fraction vertical line right parenthesis vertical line subscript 1 superscript 4 end cell row blank cell equals 1 fourth left parenthesis ln invisible function application 15 over 19 minus ln invisible function application 3 over 7 right parenthesis equals 1 fourth ln invisible function application 35 over 19 end cell end table end style

begin mathsize 20px style integral subscript 3 superscript 4 fraction numerator 5 x plus 5 over denominator x squared plus x minus 6 end fraction d x end style (23)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 5 x plus 5 over denominator x squared plus x minus 6 end fraction equals fraction numerator 5 x plus 5 over denominator left parenthesis x minus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction equals fraction numerator A over denominator x minus 2 end fraction plus fraction numerator B over denominator x plus 3 end fraction end cell row blank cell not stretchy rightwards double arrow 5 x plus 5 equals A left parenthesis x plus 3 right parenthesis plus B left parenthesis x minus 2 right parenthesis end cell row blank cell x equals 2 not stretchy rightwards double arrow A equals 3 end cell row blank cell x equals negative 3 not stretchy rightwards double arrow B equals 2 end cell row blank cell integral subscript 3 superscript 4 fraction numerator 5 x plus 5 over denominator x squared plus x minus 6 end fraction d x equals integral subscript 3 superscript 4 left parenthesis fraction numerator 3 over denominator x minus 2 end fraction plus fraction numerator 2 over denominator x plus 3 end fraction right parenthesis d x end cell row blank cell equals left parenthesis 3 ln invisible function application vertical line x minus 2 vertical line plus 2 ln invisible function application vertical line x plus 3 vertical line right parenthesis vertical line subscript 3 superscript 4 end cell row blank cell equals 3 ln invisible function application 2 plus 2 ln invisible function application 7 minus 2 ln invisible function application 6 equals ln invisible function application 98 over 9 end cell end table end style

begin mathsize 20px style integral subscript 3 superscript 4 fraction numerator 4 over denominator x cubed minus 4 x squared plus 4 x end fraction d x end style (24)

begin mathsize 20px style b b b table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 4 over denominator x cubed minus 4 x squared plus 4 x end fraction equals fraction numerator 4 over denominator x left parenthesis x minus 2 right parenthesis squared end fraction equals A over x plus fraction numerator B over denominator x minus 2 end fraction plus fraction numerator C over denominator left parenthesis x minus 2 right parenthesis squared end fraction end cell row blank cell not stretchy rightwards double arrow 4 equals A left parenthesis x minus 2 right parenthesis squared plus B x left parenthesis x minus 2 right parenthesis plus C x end cell row blank cell x equals 0 not stretchy rightwards double arrow A equals 1 end cell row blank cell x equals 2 not stretchy rightwards double arrow C equals 2 end cell row blank cell x equals 1 not stretchy rightwards double arrow 4 equals A minus B plus C not stretchy rightwards double arrow B equals negative 1 end cell row blank cell A equals integral subscript 3 superscript 4 fraction numerator 4 over denominator x cubed minus 4 x squared plus 4 x end fraction d x equals integral subscript 3 superscript 4 left parenthesis 1 over x plus fraction numerator negative 1 over denominator x minus 2 end fraction plus fraction numerator 2 over denominator left parenthesis x minus 2 right parenthesis squared end fraction right parenthesis d x end cell row blank cell equals left parenthesis ln invisible function application vertical line x vertical line minus ln invisible function application vertical line x minus 2 vertical line minus fraction numerator 2 over denominator x minus 2 end fraction right parenthesis vertical line subscript 3 superscript 4 end cell row blank cell equals left parenthesis ln invisible function application vertical line fraction numerator x over denominator x minus 2 end fraction vertical line minus fraction numerator 2 over denominator x minus 2 end fraction right parenthesis vertical line subscript 3 superscript 4 end cell row blank cell equals ln invisible function application 2 minus 1 minus ln invisible function application 3 plus 2 equals 1 plus ln invisible function application 2 over 3 end cell end table end style

أجد مساحة المنطقة المظللة في كل من التمثيلين البيانيين الآتيين:

التمثيل البياني للسؤال 25

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell A equals integral subscript 0 superscript 1 fraction numerator 1 over denominator x squared minus 5 x plus 6 end fraction d x end cell row blank cell fraction numerator 1 over denominator x squared minus 5 x plus 6 end fraction equals fraction numerator 1 over denominator left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis end fraction equals fraction numerator A over denominator x minus 3 end fraction plus fraction numerator B over denominator x minus 2 end fraction end cell row blank cell not stretchy rightwards double arrow bold 1 equals A left parenthesis x minus 2 right parenthesis plus B left parenthesis x minus 3 right parenthesis end cell row blank cell x equals 3 not stretchy rightwards double arrow A equals 1 end cell row blank cell x equals 2 not stretchy rightwards double arrow B equals negative 1 end cell row blank cell A equals integral subscript 0 superscript 1 fraction numerator 1 over denominator x squared minus 5 x plus 6 end fraction d x equals integral subscript 0 superscript 1 left parenthesis fraction numerator 1 over denominator x minus 3 end fraction plus fraction numerator negative 1 over denominator x minus 2 end fraction right parenthesis d x end cell row blank cell equals left parenthesis ln invisible function application vertical line x minus 3 vertical line minus ln invisible function application vertical line x minus 2 vertical line right parenthesis vertical line subscript 0 superscript 1 end cell row blank cell equals ln invisible function application vertical line fraction numerator x minus 3 over denominator x minus 2 end fraction vertical line vertical line subscript 0 superscript 1 end cell row blank cell equals ln invisible function application 2 minus ln invisible function application 3 over 2 equals ln invisible function application 4 over 3 end cell end table end style

التمثيل البياني للسؤال 26

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell A equals integral subscript 1 superscript 2 fraction numerator x squared plus 1 over denominator 3 x minus x squared end fraction d x end cell row blank cell fraction numerator x squared plus 1 over denominator 3 x minus x squared end fraction equals negative 1 plus fraction numerator 3 x plus 1 over denominator 3 x minus x squared end fraction end cell row blank cell fraction numerator 3 x plus 1 over denominator 3 x minus x squared end fraction equals fraction numerator 3 x plus 1 over denominator x left parenthesis 3 minus x right parenthesis end fraction equals A over x plus fraction numerator B over denominator 3 minus x end fraction end cell row blank cell not stretchy rightwards double arrow 3 x plus 1 equals A left parenthesis 3 minus x right parenthesis plus B x end cell row blank cell x equals 0 not stretchy rightwards double arrow A equals 1 third equals negative 2 plus 1 third ln invisible function application 2 plus 1 plus 10 over 3 ln invisible function application 2 end cell row blank cell x equals 3 not stretchy rightwards double arrow B equals negative 11 over 3 ln invisible function application 2 end cell row blank cell integral subscript 1 superscript 2 fraction numerator x squared plus 1 over denominator 3 x minus x squared end fraction d x equals integral subscript 1 superscript 2 left parenthesis negative 1 plus 10 over 3 plus fraction numerator 1 over denominator 3 minus x end fraction right parenthesis d x end cell end table end style

منحنى الاقترانيبين الشكل المجاور جزءاً من منحنى الاقتران: begin mathsize 20px style bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals fraction numerator bold 4 bold x bold minus bold 5 over denominator bold 2 bold x to the power of bold 2 bold minus bold 5 bold x bold minus bold 3 end fraction end style:

(27) أجد إحداثيي النقطة begin mathsize 20px style A end style.

begin mathsize 20px style f left parenthesis x right parenthesis equals 0 not stretchy rightwards double arrow 4 x minus 5 equals 0 not stretchy rightwards double arrow x equals 5 over 4 not stretchy rightwards double arrow A left parenthesis 5 over 4 comma 0 right parenthesis end style

(28) أجد مساحة المنطقة المظللة.

begin mathsize 20px style A equals integral subscript 0 superscript 5 over 4 end superscript fraction numerator 4 x minus 5 over denominator 2 x squared minus 5 x minus 3 end fraction d x equals ln invisible function application vertical line 2 x squared minus 5 x minus 3 vertical line subscript 0 superscript 5 over 4 end superscript equals ln invisible function application 49 over 8 minus ln invisible function application 3 equals ln invisible function application 49 over 24 end style

ملاحظة: البسط هو مشتقة المقام، فلا داعي لتجزئة الكسر.

أجد كلاً من التكاملات الآتية:

begin mathsize 20px style integral fraction numerator sin invisible function application x over denominator cos invisible function application x plus cos squared invisible function application x end fraction d x end style (29)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell u equals cos invisible function application x not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals negative sin invisible function application x not stretchy rightwards double arrow d x equals fraction numerator d u over denominator negative sin invisible function application x end fraction end cell row blank cell integral fraction numerator sin invisible function application x over denominator cos invisible function application x plus cos squared invisible function application x end fraction d x equals integral fraction numerator sin invisible function application x over denominator u plus u squared end fraction cross times fraction numerator d u over denominator negative sin invisible function application x end fraction equals integral fraction numerator negative 1 over denominator u plus u squared end fraction d u end cell row blank cell fraction numerator negative 1 over denominator u plus u squared end fraction equals fraction numerator negative 1 over denominator u left parenthesis 1 plus u right parenthesis end fraction equals A over u plus fraction numerator B over denominator 1 plus u end fraction end cell row blank cell not stretchy rightwards double arrow negative 1 equals A left parenthesis 1 plus u right parenthesis plus B u end cell row blank cell u equals 0 not stretchy rightwards double arrow A equals negative 1 end cell row blank cell u equals negative 1 not stretchy rightwards double arrow B equals 1 end cell row blank cell integral fraction numerator negative 1 over denominator u plus u squared end fraction d u equals integral left parenthesis fraction numerator negative 1 over denominator u end fraction plus fraction numerator 1 over denominator 1 plus u end fraction right parenthesis d u end cell row blank cell not stretchy rightwards double arrow integral fraction numerator sin invisible function application x over denominator cos invisible function application x plus cos squared invisible function application x end fraction d x equals negative ln invisible function application vertical line cos invisible function application x vertical line plus ln invisible function application vertical line 1 plus cos invisible function application x vertical line plus C end cell row blank cell space of 2em equals ln invisible function application vertical line 1 plus u vertical line plus C end cell row blank cell space of 2em fraction numerator 1 plus cos invisible function application x over denominator cos invisible function application x end fraction vertical line plus C equals ln invisible function application vertical line 1 plus sec invisible function application x ∣ plus C end cell end table end sty

begin mathsize 20px style integral fraction numerator 1 over denominator x squared plus x square root of x end fraction d x end style (30)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell u equals square root of x not stretchy rightwards double arrow u squared equals x not stretchy rightwards double arrow d x equals 2 u d u end cell row blank cell integral fraction numerator 1 over denominator x squared plus x square root of x end fraction d x equals integral fraction numerator 1 over denominator u to the power of 4 plus u cubed end fraction 2 u d u equals integral fraction numerator 2 over denominator u cubed plus u squared end fraction d u end cell row blank cell fraction numerator 2 over denominator u cubed plus u squared end fraction equals fraction numerator 2 over denominator u squared left parenthesis u plus 1 right parenthesis end fraction equals A over u plus B over u squared plus fraction numerator C over denominator u plus 1 end fraction end cell row blank cell not stretchy rightwards double arrow 2 equals A u left parenthesis u plus 1 right parenthesis plus B left parenthesis u plus 1 right parenthesis plus C u squared end cell row blank cell u equals 0 not stretchy rightwards double arrow B equals 2 end cell row blank cell u equals negative 1 not stretchy rightwards double arrow C equals 2 end cell row blank cell u equals 1 not stretchy rightwards double arrow 2 equals 2 A plus 2 B plus C not stretchy rightwards double arrow A equals negative 2 end cell row blank cell integral fraction numerator 2 over denominator u cubed plus u squared end fraction d u equals integral left parenthesis fraction numerator negative 2 over denominator u end fraction plus 2 over u squared plus fraction numerator 2 over denominator u plus 1 end fraction right parenthesis d u end cell row blank cell space of 1em equals negative 2 ln invisible function application vertical line u vertical line minus 2 over u plus 2 ln invisible function application vertical line u plus 1 vertical line plus C end cell row blank cell not stretchy rightwards double arrow integral fraction numerator 1 over denominator x squared plus x square root of x end fraction d x equals 2 ln invisible function application vertical line fraction numerator u plus 1 over denominator u end fraction vertical line minus 2 over u plus C end cell end table end style

begin mathsize 20px style integral fraction numerator e to the power of 2 x end exponent over denominator e to the power of 2 x end exponent plus 3 e to the power of x plus 2 end fraction d x end style (31)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell u equals e to the power of x not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals e to the power of x equals u not stretchy ⟹ d x equals fraction numerator d u over denominator u end fraction end cell row blank cell integral fraction numerator e to the power of 2 x end exponent over denominator e to the power of 2 x end exponent plus 3 e to the power of x plus 2 end fraction d x equals integral fraction numerator u squared over denominator u squared plus 3 u plus 2 end fraction cross times fraction numerator d u over denominator u end fraction equals integral fraction numerator u over denominator u squared plus 3 u plus 2 end fraction d u end cell row blank cell fraction numerator u over denominator u squared plus 3 u plus 2 end fraction equals fraction numerator u over denominator left parenthesis u plus 1 right parenthesis left parenthesis u plus 2 right parenthesis end fraction equals fraction numerator A over denominator u plus 1 end fraction plus fraction numerator B over denominator u plus 2 end fraction end cell row blank cell not stretchy rightwards double arrow u equals A left parenthesis u plus 2 right parenthesis plus B left parenthesis u plus 1 right parenthesis end cell row blank cell u equals negative 1 not stretchy rightwards double arrow A equals negative 1 end cell row blank cell u equals negative 2 not stretchy rightwards double arrow B equals 2 end cell row blank cell integral fraction numerator u over denominator u squared plus 3 u plus 2 end fraction d u equals integral left parenthesis fraction numerator negative 1 over denominator u plus 1 end fraction plus fraction numerator 2 over denominator u plus 2 end fraction right parenthesis d u end cell row blank cell equals negative ln invisible function application vertical line u plus 1 vertical line plus 2 ln invisible function application vertical line u plus 2 vertical line plus C end cell row blank cell not stretchy rightwards double arrow integral fraction numerator e to the power of 2 x end exponent over denominator e to the power of 2 x end exponent plus 3 e to the power of x plus 2 end fraction d x equals negative ln invisible function application left parenthesis e to the power of x plus 1 right parenthesis plus 2 ln invisible function application left parenthesis e to the power of x plus 2 right parenthesis plus C end cell end table end sty

begin mathsize 20px style integral fraction numerator cos invisible function application x over denominator sin invisible function application x left parenthesis sin squared invisible function application x minus 4 right parenthesis end fraction d x end style (32)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell u equals sin invisible function application x not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals cos invisible function application x not stretchy rightwards double arrow d x equals fraction numerator d u over denominator cos invisible function application x end fraction end cell row blank cell integral fraction numerator cos invisible function application x over denominator sin invisible function application x left parenthesis sin squared invisible function application x minus 4 right parenthesis end fraction d x equals integral fraction numerator cos invisible function application x over denominator u left parenthesis u squared minus 4 right parenthesis end fraction cross times fraction numerator d u over denominator cos invisible function application x end fraction equals integral fraction numerator 1 over denominator u left parenthesis u squared minus 4 right parenthesis end fraction d u end cell row blank cell fraction numerator 1 over denominator u left parenthesis u squared minus 4 right parenthesis end fraction equals fraction numerator 1 over denominator u left parenthesis u minus 2 right parenthesis left parenthesis u plus 2 right parenthesis end fraction equals A over u plus fraction numerator B over denominator u minus 2 end fraction plus fraction numerator C over denominator u plus 2 end fraction end cell row blank cell not stretchy rightwards double arrow 1 equals A left parenthesis u minus 2 right parenthesis left parenthesis u plus 2 right parenthesis plus B u left parenthesis u plus 2 right parenthesis plus C u left parenthesis u minus 2 right parenthesis end cell row blank cell u equals 0 not stretchy rightwards double arrow A equals negative 1 fourth end cell row blank cell u equals 2 not stretchy rightwards double arrow B equals 1 over 8 end cell row blank cell u equals negative 2 not stretchy rightwards double arrow C equals 1 over 8 end cell row blank cell integral fraction numerator 1 over denominator u left parenthesis u squared minus 4 right parenthesis end fraction d u equals integral left parenthesis fraction numerator negative 1 fourth over denominator u end fraction plus fraction numerator 1 over 8 over denominator u minus 2 end fraction plus fraction numerator 1 over 8 over denominator u plus 2 end fraction right parenthesis d u end cell row blank cell equals negative 1 fourth ln invisible function application vertical line u vertical line plus 1 over 8 ln invisible function application vertical line u minus 2 vertical line plus 1 over 8 ln invisible function application vertical line u plus 2 vertical line plus C end cell row blank cell not stretchy rightwards double arrow integral fraction numerator cos invisible function application x over denominator sin invisible function application x left parenthesis sin squared invisible function application x minus 4 right parenthesis end fraction d x end cell row blank cell equals negative 1 fourth ln invisible function application vertical line sin invisible function application x vertical line plus 1 over 8 ln invisible function application vertical line sin invisible function application x minus 2 vertical line plus 1 over 8 ln invisible function application vertical line sin invisible function application x plus 2 vertical line plus C end cell end table end style

إعداد : شبكة منهاجي التعليمية

10 / 07 / 2023

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