If $f(x) = f(a-x)$, then $\int_0^a x f(x) dx$ is equal to

The integral $\int_{-1/2}^{1/2} \left( [x] + \log \left( \frac{1+x}{1-x} \right) \right) dx$ is equal to (where $[.]$ is the greatest integer function):

$\int_3^5(x-3)^3(5-x)^5 d x=$

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:

The value of $\int_{-1}^{3} \left( \tan^{-1} \left( \frac{x}{x^2+1} \right) + \tan^{-1} \left( \frac{x^2+1}{x} \right) \right) dx$ is

$\int_{-1/2}^{1/2} (\cos x) \left[ \log \left( \frac{1-x}{1+x} \right) \right] dx = $