Question:
Evaluate $\int \mathrm{e}^{\mathrm{x}} \frac{(1-\mathrm{x})^{2}}{\left(1+\mathrm{x}^{2}\right)^{2}} \mathrm{dx}$
Solution:
$=\int e^{x} \frac{\left(1+x^{2}-2 x\right)}{\left(1+x^{2}\right)^{2}}$
$=\int e^{x} \frac{d x}{1+x^{2}}-\int \frac{2 x e^{x} d x}{\left(1+x^{2}\right)^{2}}$
$=\int e^{x}\left[\frac{1}{1+x^{2}}-\frac{2 x}{\left(1+x^{2}\right)^{2}}\right] d x \ldots \ldots\left(\int e^{x}\left(f(x)+f^{\prime}(x)\right)=e^{x} f(x)+c\right)$
$=e^{x} \frac{1}{1+x^{2}}+c$
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