$\int_{-\pi}^\pi \frac{2 x(1+\sin x)}{1+\cos ^2 x} d x=$

$\int_{\,\pi /6}^{\,\pi /3} {\,\frac{{dx}}{{1 + \sqrt {\cot x} }}} $ is equal to:

The number of elements in the set $S = \{x : x \in [0, 100] \text{ and } \int_{0}^{x} t^{2} \sin(x-t) dt = x^{2}\}$ is:

$\int_{0}^{\pi} \frac{x \, dx}{1+\cos \alpha \sin x}, (0 < \alpha < \pi)$ is equal to

Let $h(x) = \int\limits_0^x {g(t)dt}$,where $g(x)$ is a differentiable and odd function $\forall x \in R$ and $g(x)$ is periodic with period $3$.
Statement $1: h(x) + h(-x) = 0$ $\forall x \in R$
Statement $2: h(x) + h(-x) = 2 \int\limits_0^x {g(t)dt}$ $\forall x \in R$
Statement $3: h(3n) = 0$ $\forall n \in I$
Then which of the following statement$(s)$ is/are true?

Let $f:[-2, 3] \to [0, \infty)$ be a continuous function such that $f(1-x) = f(x)$ for all $x \in [-2, 3]$. If $R_1$ is the numerical value of the area of the region bounded by $y = f(x)$,$x = -2$,$x = 3$ and the $x$-axis,and $R_2 = \int_{-2}^3 x f(x) dx$,then: