I am attempting to prove that the function $\tan(x)1_{(0,\pi/2)}$ lies in $L^p$ for $p\in (0,1)$. To do this, I want to calculate the integral $\int_{0}^{\pi/2}\tan(x)^{p}\,dx$ for various $p$. However, I cannot make the substitution work properly here. I have attempted using the obvious substitution $u=\tan(x)$, but I have been unable to calculate the actual integral from that, probably due to my lack of experience with integration by substitution.
Is my choice of substitution correct, and if so, how do I proceed from here?
The sub. $u=\tan(x)$ gives $$\int_{0}^{\infty}{\frac{u^{p}}{1+u^{2}}du}=\int_{0}^{1}{\frac{u^{p}+u^{-p}}{u^{2}+1}du}\leq \int_{0}^{1}{(u^{p}+u^{-p})du}=\frac{2}{1-p^{2}}$$ whenever $p\in (0,1)$