Evaluate the following integrals: _ 0 ^1 e^x 1+e^ 2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int_{0}^\limits{1}\frac{\text{e}^\text{x}}{1+\text{e}^{2\text{x}}}\text{ dx}$

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Answer

Let $\text{e}^\text{x}=\text{t}$ Then, $\text{e}^\text{x}\text{ dx}=\text{dt}$
When $\text{e}^\text{x}=0,\text{t}=1$ and $\text{x}=1,\text{t}=\text{e}$
$\therefore\ \text{I}=\int_{0}^\limits{1}\frac{\text{e}^\text{x}}{1+\text{e}^{2\text{x}}}\text{ dx}$
$\Rightarrow\text{I}=\int_{1}^\limits{\text{e}}\frac{\text{dt}}{1+\text{t}^{2}}$
$\Rightarrow\text{I}=\big[\tan^{-1}\text{x}\big]^\text{e}_1$
$\Rightarrow\text{I}=\tan^{-1}\text{e}-\tan^{-1}1$
$\Rightarrow\text{I}=\tan^{-1}\text{e}-\frac{\pi}{4}$

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