Evaluate \[ \int\frac{4x^3+2x}{x^4+1}\,dx. \]
Solution
Split the integral: \[ \int\frac{4x^3}{x^4+1}\,dx+\int\frac{2x}{x^4+1}\,dx. \] The first is \(\ln(x^4+1)\). In the second, put \(u=x^2\), so \(du=2x\,dx\) and the denominator is \(u^2+1\). Thus \[ \boxed{\int\frac{4x^3+2x}{x^4+1}\,dx=\ln(x^4+1)+\arctan(x^2)+C.} \]