If ƒ'(x) = tan^–1(secx + tanx), $$ - {\pi \over 2} < x < {\pi \over 2}$$,
and ƒ(0) = 0, then ƒ(1) is equal to :

If $$\int {{{\cos xdx} \over {{{\sin }^3}x{{\left( {1 + {{\sin }^6}x} \right)}^{2/3}}}}} = f\left( x \right){\left( {1 + {{\sin }^6}x} \right)^{1/\lambda }} + c$$

where c is a constant of integration, then $$\lambda f\left( {{\pi \over 3}} \right)$$ is equal to

Let $$a \in \left( {0,{\pi \over 2}} \right)$$ be fixed. If the integral

$$\int {{{\tan x + \tan \alpha } \over {\tan x - \tan \alpha }}} dx$$ = A(x) cos 2$$\alpha $$ + B(x) sin 2$$\alpha $$ + C, where C is a

constant of integration, then the functions A(x) and B(x) are respectively :

The integral $$\int {{{2{x^3} - 1} \over {{x^4} + x}}} dx$$ is equal to :
(Here C is a constant of integration)