Question

Integrate with respect to $x$:
$\frac{1}{x^2+2x+5}$

Answer 1

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

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

$=\frac{dx}{(x+1{)}^2+4}$

$=\frac{dx}{(x+1{)}^2+2^2}$

$=\frac{1}{2}ta{n}^{-1}\left(\frac{x+2}{1}\right)+c$

$[\because \int \frac{dx}{x^2+a^2}=\frac{1}{a}ta{n}^{-1}\left(\frac{x}{a}\right)+c]$

$\therefore \int \frac{dx}{x^2+2x+5}=\frac{1}{2}ta{n}^{-1}\left(\frac{x+2}{1}\right)+c$