Integrate cosine function [closed]

How do you integrate this?

$$\int \cos (2x^2 )dx$$

I've tried using the substitution technique with $u=2x^2$ and $du=4xdx$ but I still encounter a problem since there is no other $x$ function in the formula. I've been stuck here for hours, would appreciate any help. Thanks.

Solution 1:

This is an example of a non-elementary integral, which cannot be solved the way you usually solve integrals. So let's use power series.

Recall that $$\cos(x) = \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}$$

Therefore, our integral can be re-expressed as

$$\int \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{4n} dx$$

Which evaluates to the following power sum

$$\sum_{n=0}^{\infty}\frac{(-1)^n}{(4n+1)((2n)!)}x^{4n+1} +C$$