A value of $$\alpha $$ such that
$$\int\limits_\alpha ^{\alpha + 1} {{{dx} \over {\left( {x + \alpha } \right)\left( {x + \alpha + 1} \right)}}} = {\log _e}\left( {{9 \over 8}} \right)$$ is :

If $$\int\limits_0^{{\pi \over 2}} {{{\cot x} \over {\cot x + \cos ecx}}} dx$$ = m($$\pi $$ + n), then m.n is equal to

Let f : R $$ \to $$ R be a continuously differentiable function such that f(2) = 6 and f'(2) = $${1 \over {48}}$$. If $$\int\limits_6^{f\left( x \right)} {4{t^3}} dt$$ = (x - 2)g(x), then $$\mathop {\lim }\limits_{x \to 2} g\left( x \right)$$ is equal to :

The integral $$\int\limits_{\pi /6}^{\pi /3} {{{\sec }^{2/3}}} x\cos e{c^{4/3}}xdx$$ is equal to :