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

1. Jul 26, 2010

### Zvaigzdute

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

3. The attempt at a solution

$$\int x^{3}/(x^{3}+1) dx$$
1. The problem statement, all variables and given/known data

2. Relevant equations

3. 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

Last edited: Jul 26, 2010
2. Jul 26, 2010

### 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.

3. Jul 27, 2010

### HallsofIvy

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