Simplifying Trigonometric Integrals

[Yae Miteo]

Homework Statement

Evaluate the integral.

Homework Equations

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

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?

 

[STEMucator]

Homework Statement

Evaluate the integral.

Homework Equations

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

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?

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

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?