# Problem with trig integral

1. Apr 13, 2014

### Yae Miteo

1. The problem statement, all variables and given/known data

Evaluate the integral.

2. Relevant equations

$$\int sin^2(\pi x) cos^5 (\pi x) dx$$

3. The attempt at a solution

I tried first by splitting the cosine up

$$\int sin^2(x) [1-cos^2(x)] cos^2(x) cos(x) dx$$ and from there use u-substitution. However, I am not sure what to substitute. Any ideas?

Last edited: Apr 13, 2014
2. Apr 13, 2014

### Zondrina

$$\int sin^2(\pi x) cos^5 (\pi x) dx$$
$$= \int sin^2(\pi x) cos^4 (\pi x) cos(\pi x) dx$$
$$= \int sin^2(\pi x) (1 - sin^2(\pi x))^2 cos(\pi x) dx$$

Since the power of sin was even and cos was odd, you should save a factor of $cos(x)$ and convert the remaining $cos^2(x)$ terms to their $1 - sin^2(x)$ equivalents.

Can you see a substitution from here that would help?