First slide

Methods of integration

Question

$\int \frac{x e^{x}}{(1 + x)^{2}} d x is equal to$

Moderate

A

$\frac{e^{x}}{x + 1} + C$
B

$e^{x} (x + 1) + C$
C

$- \frac{e^{x}}{(x + 1)^{2}} + C$
D

$\frac{e^{x}}{1 + x^{2}} + C$

Solution

$\begin{array}{l} \int e^{x} \frac{x}{(x + 1)^{2}} dx = \int e^{x} \frac{x + 1 - 1}{(x + 1)^{2}} dx \\ = \int e^{x} (\frac{1}{x + 1} + \frac{- 1}{(x + 1)^{2}}) dx \\ = e^{x} \frac{1}{x + 1} + C \end{array}$

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