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Q3  Integrate the functions x ^ 2 e ^x

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Given function is
f(x)=x^2e^x
We will use integration by parts method
\int x^2e^x= x^2.\int e^xdx - \int(\frac{d(x^2)}{dx}.\int e^x dx)dx\\ \\ \int x^2e^x = x^2.e^x- \int (2x.e^x)dx\\
Again use integration by parts in \int (2x.e^x)dx\\
\int (2x.e^x)dx = 2x.\int e^x dx - \int (\frac{d(2x)}{dx}.\int e^xdx)dx\\ \int 2x.e^x dx = 2xe^x- \int 2.e^xdx\\ \int 2x.e^x dx = 2xe^x- 2e^x
Put this value in our equation
we will get,
\int x^2.e^x dx =x^2e^x -2xe^x+ 2e^x + C\\ \int x^2.e^x dx = e^x(x^2-2x+2)+ C

Therefore, answer is e^x(x^2-2x+2)+ C
 

Posted by

Gautam harsolia

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