Let $f:[0,1] \rightarrow[0, \infty)$ be a continuous function such that $\int_0^1 f(x) d x=10$. Which of the following statements is NOT necessarily true?

$I=\int \limits_{\pi / 4}^{\pi / 3}\left(\frac{8 \sin x-\sin 2 x}{x}\right) d x$. Then

Let $I_1 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}\sin (x)dx} $ ; $I_2 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}dx} $ ; $I_3 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}(1 + x)\,dx} $

and consider the statements

$I\,:$ $I_1 < I_2$   

$II\,:$  $I_2 < I_3$ 

$III\,:$  $I_1 = I_3$

Which of the following is $(are)$ true?

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

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

If $\int_{}^{} {f(x)\,dx} = x{e^{ - \log |x|}} + f(x),$ then $f(x)$ is