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

1 Answer

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)