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

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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.

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