integrate $\int \frac{\tan \sqrt{x}}{x\sqrt{x}} $

Question

I need integrate $$\int \frac{\tan \sqrt{x}}{x\sqrt{x}} dx$$ I using $u = \sqrt{x}$, then $$2\int \frac{\tan u}{u^{2}} du$$

I try to use sustitution again, and for part, but a dont have answer.

Answer 1

As said in comments, the problem cannot be $$I=2\int \frac{ \tan (t)}{t^2}\,dt$$ which does not show analytical expression even using special function.

More than likely, they ask for $$I=2\int \frac{ \tan^{-1} (t)}{t^2}\,dt$$ which is easy to integrate by parts $$u=\tan^{-1} (t)\implies u'=\frac{dt}{t^2+1}$$ $$v'=\frac {dt} {t^2}\implies v=-\frac 1 t$$ All of that makes $$I=2\left(-\frac{\tan ^{-1}(t)}{t} +\int \frac{dt}{t \left(t^2+1\right)}\right)$$ For the last integral, partial fraction decomposition gives $$\frac{1}{t \left(t^2+1\right)}=\frac{1}{t}-\frac{t}{t^2+1}$$ which is easy to integrate.