Question

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

Answer 1

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

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

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

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

$=\frac{x^3}{3}+x-2x+2{\tan }^{-1}x$

$=\frac{x^3}{3}-x+2{\tan }^{-1}x$.