Question

What is the integral of `cos^3 x`?

Answer 1

`intcos^3x dx = sinx -sin^3x/3 + "constant"`

Explanation:

There are multiple ways to get at this integral which give alternate forms as output. A simple way to do it is to break up the integrand using

`cos^2x+sin^2x = 1`

Which gives us

`intcos^3x dx = int(1-sin^2x)cosxdx`

`= int cosxdx- int sin^2xcosxdx`

We can now do each term separately, where the first is simply:

`int cosxdx = sinx`

and the second term can be simplified by making the substitution:

`u = sinx implies du = cosxdx`

`int u^2du = (u^3)/3 = sin^3x/3`

Putting these together we get

`intcos^3x dx = sinx -sin^3x/3 + "constant"`