Let $\ln x$ denote the logarithm of $x$ with respect to the base $e$. Let $S \subset R$ be the set of all points where the function $\ln \left(x^2-1\right)$ is well-defined. Then, the number of functions $f: S \rightarrow R$ that are differentiable, satisfy $f^{\prime}(x)=\ln \left(x^2-1\right)$ for all $x \in S$ and $f(2)=0$, is

Number of values of $x$ satisfying the equation

$\int\limits_{ - \,1}^x {\,\left( {8{t^2} + \frac{{28}}{3}t + 4} \right)\,dt} $ $=$ $\frac{{\left( {{\textstyle{3 \over 2}}} \right)x + 1}}{{{{\log }_{(x + 1)}}\sqrt {x + 1} }}$ , is

Let $I = \mathop \smallint \limits_0^1 \frac{{\sin x}}{{\sqrt x }}\;dx$ and $\;J = \mathop \smallint \limits_0^1 \frac{{\cos x}}{{\sqrt x }}\;dx$ Then which one of  the following is true?

The minimum value of the function $f(x)=\int \limits_0^2 e^{|x-t|} d t$ is

Let $I_n=\int_0^{\pi / 2} x^n \cos x d x$, where $n$ is a non-negative integer. Then, $\sum \limits_{n=2}^{\infty}\left(\frac{I_n}{n !}+\frac{I_n-2}{(n-2) !}\right)$ equals

The true solution set of the inequality,

$\sqrt {5\,x\,\, - \,\,6\,\, - \,\,{x^2}} \,\, + \,\,\frac{\pi }{2}\,\,\int\limits_0^x {} $$dz > x \int\limits_0^\pi  {} sin^2 x \,dx$ is :