I would calculate (by using a series of functions probably) this integral:
$$I(x)=\int_0^x\ t^{a}\ (\sin\ t)^{b} dt $$
where $$ x\in\ [0,\frac{\pi}{2}] \ \ \ \ \| \ \ \ a,b \in\mathbb{N}$$
Using the hypergeometric functions:
$$I(x)=\int_0^x\ t^{a+1}\ (\ _0F_1(\frac{3}{2};-\frac{1}{4}t^2)\ )^b \ dt$$
But in this way I do not think it is possible to continue integration
Thanks in advance