Question #2d98d

1 Answer
Dec 19, 2017

The answer is 0.

Explanation:

Since we can pull out a^3, we can focus on

int_0^pi sin^3(2phi) sin(phi) dphi.

Using the double-angle formula for the sine, i.e., sin(2phi) = 2 sin(phi) cos(phi), this is the same as

8 int_0^pi sin^4(phi) cos^3(phi) dphi.

Now we use the identity cos^2(phi) = 1 - sin^2(phi), and arrive at

8 int_0^pi sin^4(phi) (1 - sin^2(phi)) cos(phi) dphi.

At this point we identify an inner function sin(phi), i.e., x = sin(phi) and dx = cos(phi) dphi. Therefore, we get

8 int_sin(0)^sin(phi) x^4 (1 - x^2) dx = 8 int_0^0 x^4 (1 - x^2) dx = 0.

Hope this helps :)