How to integrate $\int\frac{x}{1+x^3}dx$? How to integrate $\displaystyle \int\frac{x}{1+x^3}dx$? I tried using partial fractions and substitution but it didn't work, thanks.

Partial fractions is indeed the way to go.  If you don't want to involve complex numbers,
$1 + x^3 = (1+x)(1 - x + x^2)$ so your partial fraction decomposition will look like
$$ \frac{x}{1+x^3} = \frac{a}{1+x} + \frac{bx+c}{1-x+x^2}$$
For integrating the last term, use completing the square.
