# Definite integral of an even function

1. Oct 18, 2009

### Samuelb88

1. The problem statement, all variables and given/known data
Integrate the definite integral

$$\int_{-2}^{2}{\frac{x^2}{4+x^6} dx$$

2. Relevant equations

3. The attempt at a solution
(1) The integrand f is an even function, therefore:

$$2\int_{0}^{2}{\frac{x^2}{4+x^6} dx$$

(2) I re-expressed the denominator as:

$$2\int_{0}^{2}{\frac{x^2}{4+(x^3)^2} dx$$

$$t=x^3$$

$$\frac{1}{3}\right) dt = x^2dt$$

$$\frac{2}{3}\right) \int_{0}^{8} {\frac{1}{4+t^2} dt$$

(4) Here's where I get stuck. I can seem to make another substitution to be able to simplify the integral such that it can be evaluated or be able use integration by parts to be able to evaluate it.

Last edited: Oct 18, 2009
2. Oct 18, 2009

### Mentallic

That integral can be found on any standard integrals sheet such that:

$$\int {\frac{1}{4+t^2} dt$$

$$=\frac{1}{2}tan^{-1}(\frac{t}{2})+c$$

or are you unsatisfied with this?

3. Oct 18, 2009

### Staff: Mentor

For your last integral you need to to a trig substitution, or else know this integration formula (which can be derived by a trig substitution):
$$\int \frac{dx}{a^2 + x^2}~=~\frac{1}{a} tan^{-1}(x/a) + C$$

4. Oct 18, 2009

### Samuelb88

ahh, thanks guys. we're going to start trig. substitution next week so i guess i'm satisfied for now. :)