Using $x^{1/2} = u$ as a substitution to show that one of the following is a definite integral

Question

Integrate with respect to $x$:

$$\int\frac{1}{\sqrt{x}\sin(\sqrt x)}dx$$

or integrate with respect to $x$:

$$\int\frac{1}{\sqrt{x}\cos(\sqrt x)}dx$$

The question asks to use the substitution $u = \sqrt x$ and show that just one of the following integrals is defined.

I got up to integrating both and getting

the integral of $2/\sin(u)$ and $2/\cos(u)$ but since i got the same answers for both it seems to be wrong. Does anyone know where I've got wrong?

Thanks

Answer 1

$$ \int\frac{1}{\sqrt{x}\sin(\sqrt x)}\,dx= 2\int\frac{1}{\sin(\sqrt{x})}\frac{d}{dx}\left(\sqrt{x}\right)\,dx=\\ 2\int\frac{1}{\sin(\sqrt{x})}\,d\left(\sqrt{x}\right)=\ (u=\sqrt{x})\\ 2\int\frac{1}{\sin u}\,du. $$ Now, watch this YouTube video on how to integrate $\int\frac{1}{\sin{x}}\,dx$ (it's a long process). It should be equal to $\ln\left|\tan{\left(\frac{x}{2}\right)}\right|+C$.