Question

How do I evaluate this integral?%20%5C%2C%20dx%20" id="TexFormula1" title=" \int\limits { \frac{x}{x^4+1} } \, dx " alt=" \int\limits { \frac{x}{x^4+1} } \, dx " align="absmiddle" class="latex-formula">

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

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

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.