Evaluate $$\int_\limits2^3 x^2 d x$$ as the limit of a sum

$$\int_0^{\pi / 2} \frac{\cos x \sin x}{1+\sin x} d x$$ is equal to

If $$I_n=\int_0^{\frac{\pi}{4}} \tan ^n x d x$$, where $$n$$ is positive integer, then $$I_{10}+I_8$$ is equal to

The value of $$\int_0^{4042} \frac{\sqrt{x} d x}{\sqrt{x}+\sqrt{4042-x}}$$ is equal to