Convergence of $\int_0^{\pi/2}(\tan x)^p\,dx$ 
For what values of $p$ the integral is converge/diverge?
$$\int_{0}^{\pi/2} (\tan x)^p ~{\rm d}x$$

I tried use the fact that $\displaystyle{\tan x=\frac{\sin x}{\cos x}}$ but it didn't work.

You can use the substitution $\tan (x) = t$ to get
$$\int \limits_0^{\pi/2} (\tan(x))^p \, \mathrm{d} x = \int \limits_0^\infty \frac{t^p}{1+t^2} \, \mathrm{d} t \, .$$
Now think about how the integrand behaves for small and large values of $t$ and you should find that $-1 < p < 1$ must hold for the integral to be finite.
