Evaluate the following integrals:

Question:

Evaluate the following integrals:

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

Solution:

Let $I=\int e^{x} \frac{x+1-2}{(x+1)^{2}} d x$

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

$=\int e^{x} \frac{1}{(x+1)^{2}} d x+\int e^{x} \frac{-2}{(x+1)^{2}} d x$'

Integrating by parts

$=e^{x} \frac{1}{(x+1)^{2}}-\int e^{x} \frac{-2}{(x+1)^{2}}+\int e^{x} \frac{-2}{(x+1)^{2}}$

$=e^{x} \frac{1}{(x+1)^{2}}+c$

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