Question

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

Answer 1

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

Put $t=x-\frac{1}{x}⟹dt=(1+\frac{1}{x^2})dx$

$\int \frac{dt}{t^2+2}=\frac{1}{\sqrt{2}}{\text{tan}}^{-1}(\frac{t}{\sqrt{2}})+C=\frac{1}{\sqrt{2}}{\text{tan}}^{-1}(\frac{x^2-1}{\sqrt{2}x})+C$