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

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

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 .

##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';; 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.