Question

How do you integrate `int sin(x)^3 * cos(x) dx`?

Answer 1

Use a `u`-substitution to get `intsin^3xcosxdx=sin^4x/4+C`.

Explanation:

What we have in this integral is a function, `sinx`, and its derivative, `cosx`. That means the integral is solvable using a `u`-substitution:
Let `u=sinx->(du)/dx=cosx->du=cosxdx`
With this substitution, `intsin^3xcosxdx` becomes:
`intu^3du`

This new integral is easily evaluated using the reverse power rule:
`intu^3du=(u^(3+1))/(3+1)+C=u^4/4+C`

Because `u=sinx`, we can substitute to get a final answer of:
`intsin^3xcosxdx=sin^4x/4+C`