If $\alpha \in (2 , 3) $ then number of solution of the equation $\int\limits_0^\alpha  {}  \cos (x + \alpha^2)\, dx = \sin \,\alpha$ 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

For $x \in R$, let $f(x)=|\sin x|$ and $g(x)=\int_0^x f(t) d t .$ Let $p(x)=g(x)-\frac{2}{\pi} x$ Then

Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1, \mathrm{f}(1))$ and $(3, \mathrm{f}(3))$ make angles $\frac{\pi}{6}$ and $\frac{\pi}{4}$, respectively with positive $x$-axis. If $27 \int_1^3\left(\left(f^{\prime}(t)\right)^2+1\right) f^{\prime \prime}(t) d t=\alpha+\beta \sqrt{3} \quad$ where $\alpha, \quad \beta$ are integers, then the value of $\alpha+\beta$ equals

If $\frac{d}{{dx}}\,G\left( x \right) = \frac{{{e^{\tan \,x}}}}{x},\,x \in \left( {0,\pi /2} \right)$, then $\int\limits_{1/4}^{1/2} {\frac{2}{x}} .{e^{\tan \,\left( {\pi \,{x^2}} \right)}}dx$ is equal to

The number of continuous functions $f :\left[0, \frac{3}{2}\right] \rightarrow(0, \infty)$ satisfying the equation $4 \int \limits_0^{3 / 2} f(x) d x+125 \int \limits_0^{3 / 2} \frac{d x}{\sqrt{f(x)+x^2}}=108$ is