How to integrate this function $ x^3 \cos(2x^2-5) dx$

Question

$$\int x^3 \cos(2x^2-5)dx$$

I tried using the substitution method but to no avail.

I also tried integrating by parts but that didn't work either.

Answer 1

Hint: you don't like the $2x^2-5$. So take $u=2x^2-5$. Then $du=4xdx$, $dx=du/4x$, and $x^2=\frac{u+5}{2}$. So your integral is

$$\frac{1}{4} \int \frac{u+5}{2} \cos(u) du.$$

You can split this into two integrals, one of which is easy and the other of which requires straightforward integration by parts.

Answer 2

HINT Try by parts with $u = x^2$ and $dv = x \cos(\ldots) dx$, and then by parts again should take care of it.

Answer 3

hint: $u = 2x^2 - 5$. Can you continue?

Answer 4

Put $u=2x^2-5$,then the integral equals $\frac{5}{8}\sin(2x^2-5)+\frac{1}{8}(2x^2-5)\sin(2x^2-5)+\frac{1}{8}\cos(2x^2-5)+C$.