Evaluate: (1-x 1+x^2 )^2 e^x d x - Vidyadip

Question

Evaluate: $\int\left(\frac{1-x}{1+x^2}\right)^2 e^x d x$

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Answer

Let, $I=\int\left(\frac{1}{1+x^2}-\frac{2 x}{\left(1+x^2\right)^2}\right) e^x d x$
$\begin{array}{l}=\frac{1}{1+x^2} e^x+c \\ \left(\because \int e^x\left[f(x)+f^{\prime}(x)\right] d x=e^x f(x)+c\right) \\ \quad\left[\text { as } \frac{d}{d x}\left(\frac{1}{1+x^2}\right)=\frac{-2 x}{\left(1+x^2\right)^2}\right]\end{array}$

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