Evalute : e^x x-1 (x+1)^3 d x - Vidyadip

Question

Evalute : $\int e^x \frac{x-1}{(x+1)^3} d x$

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Answer

$
\text { Let } \begin{aligned}
I & =\int e^x \cdot \frac{x-1}{(x+1)^3} d x \\
& =\int e^x\left[\frac{(x+1)-2}{(x+1)^3}\right] d x \\
& =\int e^x\left[\frac{1}{(x+1)^2}-\frac{2}{(x+1)^3}\right] d x
\end{aligned}
$
Put $f(x)=\frac{1}{(x+1)^2} $
Then $f^{\prime}(x)=\frac{d}{d x}(x+1)^{-2}=-2(x+1)^{-3} \cdot \frac{d}{d x}(x+1)$
$
\begin{gathered}
\quad=\frac{-2}{(x+1)^3} \times(1+0)=\frac{-2}{(x+1)^3} \\
\therefore I=\int e^x\left[f(x)+f^{\prime}(x)\right] d x \\
=e^x \cdot f(x)+c=e^x \cdot \frac{1}{(x+1)^2}+c
\end{gathered}
$

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