Question

Evaluate: $\stackrel{2}{\underset{0}{\int }}\frac{e^x}{e^{2x}+1}dx$.

Answer 1

Now,

$\stackrel{2}{\underset{0}{\int }}\frac{e^x}{e^{2x}+1}dx$

$=\stackrel{2}{\underset{0}{\int }}\frac{e^x}{(e^x{)}^2+1}dx$

$=\stackrel{2}{\underset{0}{\int }}\frac{d\left(e^x\right)}{(e^x{)}^2+1}$

$={\left[{\tan }^{-1}{e}^x\right]}_{x=0}^{x=2}$

$={\tan }^{-1}{e}^2-\frac{\pi }{4}$.