Definite integral of an even function

Samuelb88

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

[tex]\int_{-2}^{2}{\frac{x^2}{4+x^6} dx[/tex]

2. Relevant equations



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

[tex]2\int_{0}^{2}{\frac{x^2}{4+x^6} dx[/tex]

(2) I re-expressed the denominator as:

[tex]2\int_{0}^{2}{\frac{x^2}{4+(x^3)^2} dx[/tex]

(3) I made the t-substitution:

[tex]t=x^3[/tex]

[tex]\frac{1}{3}\right) dt = x^2dt[/tex]

[tex]\frac{2}{3}\right) \int_{0}^{8} {\frac{1}{4+t^2} dt[/tex]

(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.

Mentallic

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

[tex]\int {\frac{1}{4+t^2} dt[/tex]

[tex]=\frac{1}{2}tan^{-1}(\frac{t}{2})+c[/tex]

or are you unsatisfied with this?

Mark44

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):
[tex]\int \frac{dx}{a^2 + x^2}~=~\frac{1}{a} tan^{-1}(x/a) + C[/tex]

Samuelb88

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