Posted by Gabe on Thursday, December 1, 2011 at 1:53am.
In exams, watch out for these freebies, definite integrals only!
If you can show that the function to be integrated is odd (i.e. f(x)=-f(x), and the limits of integration is symmetric around 0, e.g. -%pi/4 to %pi/4, etc., the answer is zero!
Try that with known odd functions, ex. y=x, y=x^3, y=sin(x), y=tan(x), etc.
When it is a product of an even function (x^2) and an odd function (sin(x)), the result is still odd: example:
(x^2)tan(x) is odd, so
∫x^2 tan(x)dx =0 if integrated from -π/4 to π/4.
But DO SHOW that the function is odd and the limits of integration are symmetric around zero before making the conclusion.
f(x)=-f(-x) ∀x => odd function