I wonder if it's possible to write the following integral in a closed form $$I(m,n)=\int_0^{\pi/2}t^{m-1}\sin^n(t)dt\quad\text{with}\quad n,m\in\mathbb{N}$$ in terms of some special function.
I be able to calculate (with Wolfram Mathematica) some of them, for small values of $n$, for example
$$I(m,1)=\int_0^{\pi/2}t^{m-1}\sin(t)dt=\frac{\left(\frac{2}{\pi }\right)^{-m-1} \, _1F_2\left(\frac{m}{2}+\frac{1}{2};\frac{3}{2},\frac{m}{2}+\frac{3}{2};-\frac{\pi ^2}{16}\right)}{m+1}$$ but not a closed form for any $n$.
Any help is welcomed.