Let $f$ be a continuous function defined on $[0,1]$ such that $\int_0^1 f^2(x) d x=\left(\int_0^1 f(x) d x\right)^2$. Then, the range of $f$

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 value of $\int_1^3 {\sqrt {3 + {x^3}} \,dx} $ lies in the interval

Let $\mathrm{a}$ and $\mathrm{b}$ be real constants such that the function $f$ defined by $f(x)=\left\{\begin{array}{cc}x^2+3 x+a & x \leq 1 \\ b x+2, & x>1\end{array}\right.$ be differentiable on $R$. Then, the value of $\int_{-2}^2 f(x) d x$ 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

The value of integral $\int_0^1 {{e^{{x^2}}}} dx$ lies in interval