Use partial fractions to integrate x^3/x^3+1

[Zvaigzdute]

Use partial fractions to integrate x^3/(x^3+1)

The Attempt at a Solution

$$\int x^{3}/(x^{3}+1) dx$$

Homework Statement

Homework Equations

The Attempt at a Solution

$$\int x^{3}/x^{3}+1 dx$$

I know that first i have to perform long division but i am at a loss how to do this

THanks

 

[hgfalling]

You can do long division, or you can just rewrite the integrand:

$$\frac{x^3}{x^3 + 1} = 1 - \frac{1}{x^3 + 1}$$

Now use partial fractions on that. Something similar comes out of long division. Then you have a big mess to slog through. The answer has two log terms and an arctan term.

 

[HallsofIvy]

If you are wondering how hgfalling got that without doing "long division", I suspect the rewrote it as
$$\frac{x^3}{x^3+ 1}= \frac{x^3+ 1- 1}{x^3+ 1}= \frac{x^3+ 1}{x^3+ 1}- \frac{1}{x+1}$$