# Trigometric for odd function

• Nov 28th 2009, 08:39 PM
Chris0724
Trigometric for odd function
For trigometric if the integrand inside the integral (i.e., the function to be integrated in theta )is ODD.

How do we approach the problem? is there a way ?
• Nov 28th 2009, 10:02 PM
simplependulum
Quote:

Originally Posted by Chris0724
For trigometric if the integrand inside the integral (i.e., the function to be integrated in theta )is ODD.

How do we approach the problem? is there a way ?

If the function can be transformed to this form :

$\displaystyle g(\theta) = \sin(a\theta)f(\cos(a\theta))$

For finding the anti-derivative , we just sub.

$\displaystyle x = \cos(a\theta) ,~~ \sin(a\theta) ~ d\theta = -\frac{1}{a}dx$

$\displaystyle I = -\frac{1}{a} \int f(x) ~dx$

We can always find out such function but hardly transform it to that form ( not impossible )

For example ,

$\displaystyle g(\theta) = \sin^{2n+1}(\theta)$

we have $\displaystyle f(\cos(\theta)) = \sin^{2n}(\theta)$

so $\displaystyle f(x) = (1-x^2)^n$