Hello
First integral can't be solved symbolically. Integral with gamma function has no antiderivative with standard elementary function or maybe known special function, no discovery yet. Use NIntegrate,a numericall integrator.
NIntegrate[Gamma[n + 1]/Gamma[k + 1]*1/Gamma[n - k + 1]*(1/10000)^k, {n, 25, 500}, {k, 2, 25}]
(* 0.0481051 *)
Second integral, you did little mistake in syntax.
Integrate[(x*y^2)*(1/(x^2 + 1)), {x, 0, 1}, {y, -3, 3}]
(* Log[512] *)
Regards Mariusz
PS: Next time post code not image.SEE