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

The value of the integral $\sum\limits_{k = 1}^n {\int_0^1 {f(k - 1 + x)\,dx} } $ is

The smallest interval $[a,\,\,b]$ such that $\int_0^1 {\frac{{dx}}{{\sqrt {1 + {x^4}} }}} \in [a,\,\,b]$ is given by

The function $L(x) = \int_1^x {\frac{{dt}}{t}} $ satisfies the equation

The number of continuous functions $f:[0,1] \rightarrow(-\infty, \infty)$ satisfying the condition $\int \limits_0^1(f(x))^2 dx =2 \int_0^1 f( x ) dx$ 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