The value of the

integral $$\int\limits_{ - 1}^1 {\log \left( {x + \sqrt {{x^2} + 1} } \right)dx} $$ is :

The value of the definite integral $$\int\limits_{\pi /24}^{5\pi /24} {{{dx} \over {1 + \root 3 \of {\tan 2x} }}} $$ is :

Let $$f:[0,\infty ) \to [0,\infty )$$ be defined as $$f(x) = \int_0^x {[y]dy} $$

where [x] is the greatest integer less than or equal to x. Which of the following is true?

If $$\int\limits_0^{100\pi } {{{{{\sin }^2}x} \over {{e^{\left( {{x \over \pi } - \left[ {{x \over \pi }} \right]} \right)}}}}dx = {{\alpha {\pi ^3}} \over {1 + 4{\pi ^2}}},\alpha \in R} $$ where [x] is the greatest integer less than or equal to x, then the value of $$\alpha$$ is :