First slide

Introduction to integration

Question

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

Easy

A

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

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

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

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

Solution

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

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