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NCERT Class XII Mathematics Chapter - - Solutions

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Exercise - 7.6
Integrate the functions in Exercises 134 to 155.
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Question : 136 of 271
Marks: +1, -0
x2exx^2e^x
Solution:  
Let I=∫x2exdxI=\int x^2 e^x dx
=x2∫exdx−∫(ddxx2⋅∫exdx)dx=x^2 \int e^x dx - \int \left( \frac{d}{dx} x^2 \cdot \int e^x dx \right) dx
=x2ex−[∫2xexdx]+C=x^2 e^x - \left[ \int 2x e^x dx \right] + C
=x2ex−2∫xexdx+C=x^2 e^x - 2 \int x e^x dx + C
=x2ex−2[x∫exdx−∫(ddxx⋅∫exdx)dx]+C=x^2 e^x - 2 \left[ x \int e^x dx - \int \left( \frac{d}{dx} x \cdot \int e^x dx \right) dx \right] + C
=x2ex−2[xex−∫exdx]+C=x^2 e^x - 2 \left[ x e^x - \int e^x dx \right] + C
=x2ex−2xex+2ex+C=x^2 e^x - 2 x e^x + 2 e^x + C
=ex(x2−2x+2)+C=e^x (x^2 - 2x + 2) + C
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