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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