حمّل تطبيق منهاجي الجديد

منهاجي صار أسرع من خلال التطبيق

  أتحقق من فهمي

أتحقق من فهمي

تكامل اقترانات خاصة

تكامل الاقتران الأسي الطبيعي، واقتران الجيب، واقتران جيب التمام

أتحقق من فهمي صفحة (43): 

أجد كلاً من التكاملات الآتية:

begin mathsize 20px style integral left parenthesis 5 x squared plus 7 e to the power of x right parenthesis d x end style (a)

begin mathsize 20px style integral left parenthesis 5 x squared plus 7 e to the power of x right parenthesis d x equals 5 over 3 x cubed plus 7 e to the power of x plus C end style

begin mathsize 20px style integral left parenthesis 9 cos invisible function application x plus 4 over x cubed right parenthesis d x end style (b)

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 integral left parenthesis 9 cos invisible function application x plus 4 over x cubed right parenthesis d x end cell cell equals integral left parenthesis 9 cos invisible function application x plus 4 x to the power of negative 3 end exponent right parenthesis d t end cell row blank cell equals 9 sin invisible function application x minus 2 x to the power of negative 2 end exponent plus C end cell row blank cell equals 9 sin invisible function application x minus 2 over x squared plus C end cell end table end style

begin mathsize 20px style integral left parenthesis cube root of x minus sin invisible function application x right parenthesis d x end style (c)

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 integral left parenthesis cube root of x minus sin invisible function application x right parenthesis d x end cell cell equals integral left parenthesis x to the power of 1 third end exponent minus sin invisible function application x right parenthesis d x end cell row blank cell equals 3 over 4 x to the power of 4 over 3 end exponent plus cos invisible function application x plus C end cell end table end style


تكامل الاقتران اللوغريتمي الطبيعي

أتحقق من فهمي صفحة (45):

أجد كلاً من التكاملات الآتية:

begin mathsize 20px style integral left parenthesis 1 over x plus 8 e to the power of x right parenthesis d x end style (a)

begin mathsize 20px style integral left parenthesis 1 over x plus 8 e to the power of x right parenthesis d x equals ln invisible function application vertical line x vertical line plus 8 e to the power of x plus C end style

begin mathsize 20px style integral left parenthesis sin invisible function application x minus 5 over x right parenthesis d x end style (b)

begin mathsize 20px style integral left parenthesis sin invisible function application x minus 5 over x right parenthesis d x equals negative cos invisible function application x minus 5 ln invisible function application vertical line x vertical line plus C end style

begin mathsize 20px style integral fraction numerator x squared minus 7 x plus 2 over denominator x squared end fraction d x end style (c)

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 integral fraction numerator x squared minus 7 x plus 2 over denominator x squared end fraction d x end cell cell equals integral left parenthesis x squared over x squared minus fraction numerator 7 x over denominator x squared end fraction plus 2 over x squared right parenthesis d x end cell row blank cell equals integral left parenthesis 1 minus 7 over x plus 2 x to the power of negative 2 end exponent right parenthesis d x end cell row blank cell equals x minus 7 ln invisible function application vertical line x vertical line minus x to the power of negative 1 end exponent plus C end cell row blank cell equals x minus 7 ln invisible function application vertical line x vertical line minus 1 over x plus C end cell end table end style


تكامل اقترانات أساسية في صورة f(ax+b)

أتحقق من فهمي صفحة (47):

أجد كلاً من التكاملات الآتية:

begin mathsize 20px style integral left parenthesis 7 x minus 5 right parenthesis to the power of 6 d x end style (a)

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 integral left parenthesis 7 x minus 5 right parenthesis to the power of 6 d x end cell cell equals 1 over 7 cross times 1 over 7 left parenthesis 7 x minus 5 right parenthesis to the power of 7 plus C end cell row blank cell equals 1 over 49 left parenthesis 7 x minus 5 right parenthesis to the power of 7 plus C end cell end table end style

begin mathsize 20px style integral square root of 2 x plus 1 end root d x end style (b)

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 integral square root of 2 x plus 1 end root d x end cell cell equals integral left parenthesis 2 x plus 1 right parenthesis to the power of 1 half end exponent d x end cell row blank cell equals 1 half cross times 2 over 3 left parenthesis 2 x plus 1 right parenthesis to the power of 3 over 2 end exponent plus C end cell row blank cell equals 1 third left parenthesis 2 x plus 1 right parenthesis to the power of 3 over 2 end exponent plus C end cell end table end style

begin mathsize 20px style integral 4 cos invisible function application left parenthesis 3 x minus 7 right parenthesis d x end style (c)

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 integral 4 cos invisible function application left parenthesis 3 x minus 7 right parenthesis d x end cell cell equals 1 third cross times 4 sin invisible function application left parenthesis 3 x minus 7 right parenthesis plus C end cell row blank cell equals 4 over 3 sin invisible function application left parenthesis 3 x minus 7 right parenthesis plus C end cell end table end style

begin mathsize 20px style integral left parenthesis sin invisible function application 5 x plus e to the power of 2 x end exponent right parenthesis d x end style (d)

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 integral left parenthesis sin invisible function application 5 x plus e to the power of 2 x end exponent right parenthesis d x end cell cell equals 1 fifth x minus cos invisible function application 5 x plus 1 half e to the power of 2 x end exponent plus C end cell row blank cell equals negative 1 fifth cos invisible function application 5 x plus 1 half e to the power of 2 x end exponent plus C end cell end table end style

begin mathsize 20px style integral left parenthesis 6 x squared minus 3 e to the power of 7 x plus 1 end exponent right parenthesis d x end style (e)

begin mathsize 20px style integral left parenthesis 6 x squared minus 3 e to the power of 7 x plus 1 end exponent right parenthesis d x equals 2 x cubed minus 3 over 7 e to the power of 7 x plus 1 end exponent plus C end style

begin mathsize 20px style integral fraction numerator 5 over denominator 3 x plus 2 end fraction d x end style (f)

begin mathsize 20px style integral fraction numerator 5 over denominator 3 x plus 2 end fraction d x equals 5 over 3 ln invisible function application vertical line 3 x plus 2 vertical line plus C end style

أتحقق من فهمي صفحة (49):

سكان: أشارت دراسة إلى أن عدد السكان في إحدى القرى يتغير سنوياً بمعدل يمكن نمذجته begin mathsize 20px style P to the power of straight prime left parenthesis t right parenthesis equals 105 e to the power of 0.03 t end exponent end style، حيث begin mathsize 20px style t end style عدد السنوات منذ عام 2010 م، و begin mathsize 20px style P left parenthesis t right parenthesis end style عدد السكان. أجد عدد سكان القرية عام 2020 م، علماً بأن عدد سكانها عام 2010 م هو 3500 شخص.

أولاً نجد تكامل الاقتران begin mathsize 20px style P to the power of straight prime left parenthesis t right parenthesis end style

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 P left parenthesis t right parenthesis equals integral 105 e to the power of 0.03 t end exponent d t end cell cell equals fraction numerator 105 over denominator 0.03 end fraction e to the power of 0.03 t end exponent plus C end cell row blank cell equals 3500 e to the power of 0.03 t end exponent plus C end cell end table end style

ثانياً، نجد ثابت التكامل C: 

بما أن عدد سكان المدينة عام 2010 هو 3500 شخص إذن begin mathsize 20px style P left parenthesis 0 right parenthesis equals 3500 end style

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 P left parenthesis t right parenthesis equals 3500 e to the power of 0.03 t end exponent plus C end cell row blank cell P left parenthesis 0 right parenthesis equals 3500 e to the power of 0 plus C end cell row blank cell 3500 equals 3500 plus C end cell row blank cell C equals 0 end cell row blank cell P left parenthesis t right parenthesis equals 3500 e to the power of 0.03 t end exponent end cell end table end style

ثالثاً، نجد سكان المدينة عام 2020 (أي بعد 10 سنوات): 

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 P left parenthesis 10 right parenthesis end cell cell equals 3500 e to the power of 0.03 left parenthesis 10 right parenthesis end exponent end cell row blank cell almost equal to 4725 end cell end table end style

إذن، عدد سكان المدينة عام 2020 هو 4725 ساكناً.


تكامل اقترانات في صورة begin mathsize 20px style fraction numerator f to the power of straight prime left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis end fraction end style

أتحقق من فهمي صفحة (50): 

أجد كلاً من التكاملات الآتية:

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

begin mathsize 20px style integral fraction numerator 2 x plus 3 over denominator x squared plus 3 x end fraction d x equals ln invisible function application vertical line x squared plus 3 x vertical line plus C end style

begin mathsize 20px style integral fraction numerator 9 x squared over denominator x cubed plus 8 end fraction d x end style (b)

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 integral fraction numerator 9 x squared over denominator x cubed plus 8 end fraction d x end cell cell equals integral fraction numerator 3 left parenthesis 3 x squared right parenthesis over denominator x cubed plus 8 end fraction d x end cell row blank cell equals 3 integral fraction numerator 3 x squared over denominator x cubed plus 8 end fraction d x end cell row blank cell equals 3 ln invisible function application vertical line x cubed plus 8 vertical line plus C end cell end table end style

begin mathsize 20px style integral fraction numerator x plus 1 over denominator 4 x squared plus 8 x end fraction d x end style (c)

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 integral fraction numerator x plus 1 over denominator 4 x squared plus 8 x end fraction d x end cell cell equals integral fraction numerator 1 over 8 left parenthesis 8 x plus 8 right parenthesis over denominator 4 x squared plus 8 x end fraction d x end cell row blank cell equals 1 over 8 integral fraction numerator 8 x plus 8 over denominator 4 x squared plus 8 x end fraction d x end cell row blank cell equals 1 over 8 ln invisible function application vertical line 4 x squared plus 8 x vertical line plus C end cell end table end style

begin mathsize 20px style integral fraction numerator e to the power of 3 x end exponent over denominator e to the power of 3 x end exponent plus 5 end fraction d x end style (d)

begin mathsize 20px style integral fraction numerator e to the power of 3 x end exponent over denominator e to the power of 3 x end exponent plus 5 end fraction d x equals integral fraction numerator 1 third left parenthesis 3 e to the power of 3 x end exponent right parenthesis over denominator e to the power of 3 x end exponent plus 5 end fraction d x equals 1 third ln invisible function application vertical line e to the power of 3 x end exponent plus 5 vertical line plus C end style


التكاملات المحدودة للاقترانات الخاصة

أتحقق من فهمي صفحة (51): 

أجد كلاً من التكاملات الآتية:

begin mathsize 20px style integral subscript 0 superscript 2 left parenthesis 4 e to the power of 2 x end exponent plus 7 right parenthesis d x end style (a)

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 integral subscript 0 superscript 2 left parenthesis 4 e to the power of 2 x end exponent plus 7 right parenthesis d x end cell cell equals left parenthesis 2 e to the power of 2 x end exponent plus 7 x right parenthesis vertical line subscript 0 superscript 2 end cell row blank cell equals left parenthesis 2 e to the power of 2 left parenthesis 2 right parenthesis end exponent plus 7 left parenthesis 2 right parenthesis right parenthesis minus left parenthesis 2 e to the power of 2 left parenthesis 0 right parenthesis end exponent plus 7 left parenthesis 0 right parenthesis right parenthesis end cell row blank cell equals 2 e to the power of 4 plus 12 end cell end table end style

begin mathsize 20px style integral subscript 0 superscript 4 fraction numerator 1 over denominator square root of 6 x plus 1 end root end fraction d x end style (b)

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 integral subscript 0 superscript 4 fraction numerator 1 over denominator square root of 6 x plus 1 end root end fraction d x end cell cell equals integral subscript 0 superscript 4 left parenthesis 6 x plus 1 right parenthesis to the power of negative 1 half end exponent d x end cell row blank cell equals 1 over 6 cross times 2 left parenthesis 6 x plus 1 right parenthesis to the power of 1 half end exponent vertical line subscript 0 superscript 4 end cell row blank cell equals 1 third square root of 6 x plus 1 end root vertical line subscript 0 superscript 4 end cell row blank cell equals left parenthesis 1 third square root of 6 left parenthesis 4 right parenthesis plus 1 end root right parenthesis minus left parenthesis 1 third square root of 6 left parenthesis 0 right parenthesis plus 1 end root right parenthesis end cell row blank cell equals 4 over 3 end cell end table end style

begin mathsize 20px style integral subscript 0 superscript 4 fraction numerator 8 x over denominator x squared plus 1 end fraction d x end style (c)

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 integral subscript 0 superscript 4 fraction numerator 8 x over denominator x squared plus 1 end fraction d x end cell cell equals integral subscript 0 superscript 4 fraction numerator 4 left parenthesis 2 x right parenthesis over denominator x squared plus 1 end fraction d x end cell row blank cell equals 4 integral subscript 0 superscript 4 fraction numerator left parenthesis 2 x right parenthesis over denominator x squared plus 1 end fraction d x end cell row blank cell equals 4 ln invisible function application vertical line x squared plus 1 vertical line vertical line subscript 0 superscript 4 end cell row blank cell equals left parenthesis 4 ln invisible function application vertical line left parenthesis 4 right parenthesis squared plus 1 vertical line right parenthesis minus left parenthesis 4 ln invisible function application vertical line left parenthesis 0 right parenthesis squared plus 1 vertical line right parenthesis end cell row blank cell equals 4 ln invisible function application 17 end cell end table end style

إعداد : شبكة منهاجي التعليمية

10 / 07 / 2023

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