First slide

Methods of integration

Question

$\int \frac{x . e^{x}}{{(2 + x)}^{3}} d x =$

Easy

A

$\frac{e^{x}}{{(2 + x)}^{3}} + c$
B

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

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

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

Solution

$\int \frac{x . e^{x}}{{(2 + x)}^{3}} d x =$ $\int e^{x} [\frac{(2 + x - 2)}{{(2 + x)}^{3}}] d x$
$= \int e^{x} [\frac{1}{{(2 + x)}^{2}} - \frac{2}{{(2 + x)}^{3}}] d x$

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

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