The value of $\int\limits_0^{\frac{\pi }{2}} {\sin 8x \cot x \, dx} + \int\limits_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\ln \left( {\frac{{1 - \sin x}}{{1 + \sin x}}} \right)dx}$ is equal to

Let $f(x)=x+\frac{a}{\pi^2-4} \sin x+\frac{b}{\pi^2-4} \cos x$ for $x \in R$ be a function which satisfies $f(x)=x+\int \limits_0^{\pi / 2} \sin (x+y) f(y) d y$. Then $(a+b)$ is equal to $............$

Let $I_n = \int_0^1 (\log x)^n dx$,where $n$ is a non-negative integer. Then,$I_{2011} + 2011 I_{2010}$ is equal to

Let $f: R \rightarrow R$ be a function defined by $f(x)=\begin{cases} [x], & x \leq 2 \\ 0, & x>2 \end{cases}$,where $[x]$ is the greatest integer less than or equal to $x$. If $I=\int_{-1}^2 \frac{x f(x^2)}{2+f(x+1)} dx$,then the value of $(4I-1)$ is

Let $f(\theta) = \sin \theta + \int_{-\pi / 2}^{\pi / 2} (\sin \theta + t \cos \theta) f(t) dt$. Then the value of $\left| \int_{0}^{\pi / 2} f(\theta) d\theta \right|$ is

If $A_n = \int_{0}^{\pi /2} \frac{\sin((2n-1)x)}{\sin x} dx$ and $B_n = \int_{0}^{\pi /2} \left( \frac{\sin(nx)}{\sin x} \right)^2 dx$ for $n \in N$,then: