Integral question - $\int\frac{\sin\sqrt{x}}{\sqrt{x}}dx$

Question

This is the integral : $$\int\frac{\sin\sqrt{x}}{\sqrt{x}}dx$$

I thinking about put universal identity but not sure, I know that $\sin(x) = \dfrac{2t}{1+t^2}$.

But what about the square root? Instead of $t$, I need to write $\sqrt{t}$? like $\dfrac{2\sqrt{t}}{1+\sqrt{t}^2}$?

Thanks!

Answer 1

Let $\sqrt{x} = t \implies x = t^2 \implies dx = 2tdt$. We then get $$\int \dfrac{\sin(\sqrt{x})}{\sqrt{x}}dx = \int \dfrac{\sin(t)}{t}2tdt = 2 \int \sin(t) dt$$ I trust you can take it from here.