Question

What is the integral of `x^3 cos(x^2) dx`?

Answer 1

`1/2x^2sin(x^2)+1/2cos(x^2) +C`

Explanation:

We can't just integrate straight away, so we try substitution.

While trying substitution, we observe that we could integrate `cos(x^2) x dx` by substitution.

So, let's split the integrand and use integration by parts.

`int x^3 cos(x^2) dx = int x^2 cos(x^2) x dx`

(We notice that we could rewrite this as `int u cos(u) 1/2du`, but we don't see how to integrate that, so we'll continue with parts for now.)

`int x^2 cos(x^2) x dx`

Let `u = x^2` and `dv = cos(x^2) x dx`.

Clearly `du = 2x dx`, and

we can integrate `dv` by substitution to get.

`1/2sin(x^2)`.

`uv-int vdu = 1/2x^2sin(x^2)-intxsin(x^2)dx`

Integrate by substitution agan to finish.

`int x^3 cos(x^2) dx = 1/2x^2sin(x^2)+1/2cos(x^2) +C`

Check the answer by differentiating.