Integral $\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$ Evaluate:
$$\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$$
Using real methods only.
I am not sure what to do.
I tried finding a power series, which was too ugly.
I just need some hints, not an answer to do this integral, this is from the MIT Integration bee 2012.

It is known that : $$\int_a^b\frac{f(x)}{f(a+b-x)+f(x)}dx=\frac{a+b}{2}$$You should remember this. After making sure that this is the case, you get the answer as $1005$.
