Help me with this Integral problem?

Question

int x cos (x^2) dx
shall i use substitution method or integral by parts?
and it's not cos^2 x, it's cos (x^2) so a little bit confusing.
i used substitution and product rule i got

x sin x^2 + (x^2)/2 cos x^2

is this correct? Thanks!

Answer 1

Here's what I have (I'll use S ans the integration sign):

S xcos(x^2) dx

u = x^2
du = 2x dx

S xcos(x^2) dx = (1/2) S cos(u) du
= (1/2)sin(u) + C
= (1/2)sin(x^2) + C

Substitution works better here, since there's that x^2 term in the cos and an extra x term outside the cos. If you take the derivative of the answer you get, you should get the integral you started with (good way of checking your work in the end). Hope this helps!