Question

${\int }_{-1}^1\frac{dx}{x^2+2x+5}$

Answer 1

${\int }_{-1}^1\frac{dx}{x^2+zx+5}$

${\int }_{-1}^1\frac{1}{(x+1{)}^2+4}dx$

$\frac{1}{2}{\tan }^{-1}\frac{(x+1)}{2}{|}_{-1}^1$

$\frac{1}{2}{\tan }^{-1}\left(\frac{2}{2}\right)-\frac{1}{2}{\tan }^{-1}\left(\frac{-1+1}{2}\right)$

$\frac{1}{2}{\tan }^{-1}1-\frac{1}{2}{\tan }^{-1}0$

$\frac{1}{2}\left(\frac{\pi }{4}\right)-\frac{1}{2}\left(0\right)\Rightarrow \frac{\pi }{8}-0=\frac{\pi }{8}$