First slide

Introduction to integration

Question

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

Easy

A

$e^{x} {(1 + x^{2})}^{2}$ +c
B

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

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

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

Solution

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

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