Evaluate the following integrals:

Question:

Evaluate the following integrals:

Solution:

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

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

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

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

Using integration by parts,

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

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

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