Question

Evaluate:$\int \frac{x^2+1}{x^4+1}dx$

Answer 1

$\int \frac{x^2+1}{x^4+1}dx$

$=\int \frac{1+1/x^2}{x^2+1/x^2}dx$.............dividing by $x^2$

$=\int \frac{d(x-1/x)}{(x-1/x{)}^2+2}$

$=\int \frac{d(x-1/x)}{(x-1/x{)}^2+(\sqrt{2}{)}^2}$

$=\frac{1}{\sqrt{2}}{\tan }^{-1}\left(\frac{x-1/x}{\sqrt{2}}\right)+c$