Question

How do you find the integral of `x^3 cos(x^2) dx`?

Answer 1

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

Explanation:

We set `t=x^2` hence `dt=2xdx` hence we have that

`int x^3cosx^2dx=int 1/2*x^2 cos(x^2) 2xdx=int1/2*tcostdt=1/2inttcostdt=1/2intt(sint)'dt=1/2[tsint]-1/2intt'sintdt=1/2tsint-1/2intsintdt=1/2tsint+1/2cost`

Hence

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