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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.

Sebastiano
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starglitter
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1 Answers1

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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

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

Prometheus
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    +1 : Nice answer, which begs the question: what specific training or book or class teaches that the original problem is a non-elementary integral? – user2661923 Nov 26 '21 at 07:24
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    I don't remember, I just know that. Usually when you have some special function like $\cos(x)$, $e^x$ or $\ln(x)$, and the function has an argument that is not a linear expression, but there is no x terms adjacent, then it is very likely that the integral is non-elementary. Another example is $\int e^{x^2} dx$. This doesn't cover almost every possible case (for the exponential function, having a ln expression as an argument works fine, but in general, this rarely happens) – Prometheus Nov 26 '21 at 07:27
  • @user2661923 "Which specific training of book" ... see https://math.stackexchange.com/q/155/442 – GEdgar Nov 26 '21 at 10:37