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Questions  >>  CBSE XII  >>  Math  >>  Integrals
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Integrate the function $\;\int\;\frac{xe^x}{(1+x)^2}$

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  • $\int e^x\{f(x)+f'(x)\}dx=e^xf(x)+c.$
Given $I=\int\frac {xe^x}{(1+x)^2}dx=\int e^x\begin{bmatrix}\frac{x}{(1+x)^2}\end{bmatrix}dx.$.
 
Add and subtract 1 to the numerator,
 
$I=\int e^x\begin{bmatrix}\frac{x+1-1}{(1+x)^2}\end{bmatrix}dx.$
 
Now separating the terms we get,
 
$I=\int e^x\begin{bmatrix}\frac{1}{(1+x)}-\frac{1}{(1+x)^2}\end{bmatrix}dx.$
 
Clearly here f(x)=$\frac{1}{(1+x)}$ and $f'(x)=\frac{-1}{(1+x)^2}$.
 
So $\int e^x\begin{bmatrix}\frac{1}{1+x}-\frac{1}{(1+x)^2}\end{bmatrix}=e^x\big(\frac{1}{1+x}\big)+c.$

 

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