If $$\int_0^x {f\left( t \right)dt = x + \int_x^1 {t\,\,f\left( t \right)\,\,dt,} } $$ then the value of $$f(1)$$ is

The value of $$\int\limits_\pi ^{2\pi } {\left[ {2\,\sin x} \right]\,dx} $$ where [ . ] represents the greatest integer function is

If $$f\left( x \right)\,\,\, = \,\,\,A\sin \left( {{{\pi x} \over 2}} \right)\,\,\, + \,\,\,B,\,\,\,f'\left( {{1 \over 2}} \right) = \sqrt 2 $$ and
$$\int\limits_0^1 {f\left( x \right)dx = {{2A} \over \pi },} $$ then constants $$A$$ and $$B$$ are

The value of $$\int\limits_0^{\pi /2} {{{dx} \over {1 + {{\tan }^3}\,x}}} $$ is