Let $\frac{d}{{dx}}F(x) = \left( {\frac{{{e^{\sin x}}}}{x}} \right)\,;\,x > 0$. If $\int_{\,1}^{\,4} {\frac{3}{x}{e^{\sin {x^3}}}dx = F(k) - F(1)} $, then one of the possible value of $k$, is

The number of continuous functions $f:[0,1] \rightarrow R$ that satisfy $\int \limits_0^1 x f(x) d x=\frac{1}{3}+\frac{1}{4} \int \limits_0^1(f(x))^2 d x$ is

The value of integral $\int_0^1 {\frac{{{x^b} - 1}}{{\log x}}} \,dx$ is

Let $f:[0,1] \rightarrow[0,1]$ be a continuous function such that $x^2+(f(x))^2 \leq 1$ for all $x \in[0,1]$ and $\int_0^1 f(x) d x=\frac{\pi}{4}$ Then, $\int_{\frac{1}{2}}^{\frac{1}{\sqrt{2}}} \frac{f(x)}{1-x^2} d x$ equals

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