Equality of two trigonometric integrals on [0,1]

Question

I need to show, that :

$$\int_0^1 \cos(x^2)~\mathrm{d}x = \frac{1}{2} \int_0^1 \frac{\cos x}{\sqrt x}~\mathrm{d}x$$

But frankly I cannot see way to solve it. The right-side integral is improper and as far I know both of them don't have the elementary antiderivatives.

Thanks for help!

Answer 1

Set $t=x^2$. Then $x=\sqrt{t}$. $dx=\frac{1}{2\sqrt{t}}dt$

Then change the bounds: if x=0, t=0, if x=1, t=1. So the two integrals are equal. The name of the variable does not chage the things.