Problem

Use integration by parts to find an expression for

taking u = (sin(x))⁴ and dv = sin(x) dx.

Solution

Work out the terms du and v.

Substitute into the integration by parts formula.

The "u times v" part of the expression can be evaluated at the endpoints of the integral.

We now have an expression for the original integral, but so far it isn't too useful. Let's try using a trigonometric identity, replacing the cos²(x) factor in the rightmost integrand with 1 – sin²(x).

The leftmost and rightmost parts of this equation both contain the original integral. Take these two parts and solve for the integral:

(*)

How has this gotten us any closer to a solution?

We still need to solve an integral, but this time it involves a lower power of sin. If we continue the integration by parts process on this new integral, we hope to reduce it to a final form that we can evaluate.

This time, let u = sin²(x) and dv = sin(x) dx. We get

When evaluate at the endpoints of integration, –cos(x) sin²(x) equals

As before, substitute the identity 1 – sin²(x) for cos²(x):

Solve for the integral.

Return to the equation (*) that expresses the original integral in terms of an integral of a lower power of sin. Now we can substitute for that lower power of sin to get a solution.

Note that we could have found the integral of any positive power m of sin, and equation (*) would give us an expression for the integral of sin^m(x) in terms of sin^m–2(x).