How to integrate $\int x \cos \frac{1}{x^{2}} $

Question

I am facing problems regarding integration of the cosine term here...

$$\int x \cos \frac{1}{x^{2}} $$

Answer 1

Hint.

$$\frac{1}{x^2} = y ~~~~~~~~~~~ dy = -2y\sqrt{y}\ dx$$

Hence

$$\int \frac{1}{\sqrt{y}} \cos(y) \left(-\frac{dy}{2y\sqrt{y}}\right) = -\frac{1}{2}\int \frac{\cos(y)}{y^2}\ dy$$

Then you can do a by parts integration.

Spoiler

I hope you studied special function since the final result contains one of them, precisely the so called "Sine Integral". The final solution is:

$$\frac{1}{2}\text{Si}\left(\frac{1}{x^2}\right)+\frac{1}{2} x^2 \cos \left(\frac{1}{x^2}\right)$$

Where

$$\text{Si}(x) = \int \frac{\sin(x)}{x}\ dx$$

Answer 2

Continuing Von's point, you may want to consider the series expansion of the integral at x=$\infty$. It will give you more sense of what's happening.