If $$\int_\limits0^\pi \log (\sin x) d x=8 k$$, then $$\int_\limits0^{\frac{\pi}{4}} \log (1+\tan x) d x$$ is equal to

If $$\int_\limits0^1 x^m(1-x)^n d x=k \int_\limits0^1 x^n(1-x)^m d x$$, then the value of $k$ equals

If $$\int_0^a {{{dx} \over {4 + {x^2}}} = {\pi \over 8}} $$, then the value of a is equal to

$$\int_1^2 {{{{x^3} - 1} \over {{x^2}}}} $$ is equal to