Mathematics

# Evaluate the following integral:$\displaystyle \int { \cfrac { { e }^{ x } }{ 1+{ e }^{ 2x } } } dx$

##### SOLUTION
$\displaystyle\int{\dfrac{{e}^{x}dx}{1+{\left({e}^{x}\right)}^{2}}}$

Let ${e}^{x}=\tan{\theta}\Rightarrow\,{e}^{x}dx={\sec}^{2}{\theta}d\theta$

$=\displaystyle\int{\dfrac{{\sec}^{2}{\theta}d\theta}{1+{\left(\tan{\theta}\right)}^{2}}}$

$=\displaystyle\int{\dfrac{{\sec}^{2}{\theta}d\theta}{1+{\tan}^{2}{\theta}}}$

$=\displaystyle\int{\dfrac{{\sec}^{2}{\theta}d\theta}{{\sec}^{2}{\theta}}}$

$=\displaystyle\int{d\theta}$

$=\theta+c$

$={\tan}^{-1}{\left({e}^{x}\right)}+c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
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