$\int_{ - 1/2}^{1/2} {\cos x \ln \left( \frac{1 + x}{1 - x} \right) dx}$ is equal to

If ${u_n} = \int_0^{\pi /4} {{\tan ^n}x\,dx,} $ then ${u_n} + {u_{n - 2}} = $

Let ${I_1} = \int_a^{\pi - a} {xf(\sin x)dx}$ and ${I_2} = \int_a^{\pi - a} {f(\sin x)dx}$,then ${I_2}$ is equal to

$\int_{0}^{1} x(1 - x)^{5} dx = . . . . . .$

If $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96 x^2 \cos^2 x}{1+e^x} dx = \pi(\alpha \pi^2 + \beta)$,where $\alpha, \beta \in \mathbb{Z}$,then $(\alpha + \beta)^2$ equals:

Let $g(x) = \int_{0}^{x} f(t) dt$,where $f$ is a continuous function in $[0, 3]$ such that $\frac{1}{3} \leq f(t) \leq 1$ for all $t \in [0, 1]$ and $0 \leq f(t) \leq \frac{1}{2}$ for all $t \in (1, 3]$. The largest possible interval in which $g(3)$ lies is: