Question

${\int }^{}\frac{dx}{x^2+4x+5}dx$

Answer 1

$\int \frac{dx}{x^2+4x+5}$

$\int \frac{dx}{(x+2{)}^2+12}$

Let $x+2=\tan \theta$

$dx={\sec }^2\theta d\theta$

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

$=\int \frac{{\sec }^2\theta }{{\sec }^2\theta }d\theta =\int d\theta$

$=\theta +c$

$={\tan }^{-1}(x+2)+c$