Question

How do I evaluate this integral?
`\int\limits { (x)/(x^4+1) } \, dx`

Answer 1

One thing you could try is to set
`x=\sqrt y`. This makes
`y=x^2`, so that
`\mathrm dy=2x\,\mathrm dx`, and
`y^2=x^4`. So the integral is


`\displaystyle\int\frac x{x^4+1}\,\mathrm dx=\frac12\int(\mathrm dy)/(y^2+1)=\frac12\arctan y+C=\frac12\arctan(x^2)+C`

A "trickier" way to do it is to write


`x^4+1=(x^2+\sqrt2x+1)(x^2-\sqrt2x+1)`

so you could decompose the integrand into partial fractions. But that's more work than needed.