The values of $\alpha$ which satisfy $\int_{\pi /2}^{\alpha} \sin x \, dx = \sin 2\alpha$,where $\alpha \in [0, 2\pi]$,are equal to:

For $x, t \in R$,let $p_t(x) = (\sin t) x^2 - (2 \cos t) x + \sin t$ be a family of quadratic polynomials in $x$ with variable coefficients. Let $A(t) = \int_0^1 p_t(x) dx$. Which of the following statements are true?
$I$. $A(t) < 0$ for all $t$.
$II$. $A(t)$ has infinitely many critical points.
$III$. $A(t) = 0$ for infinitely many $t$.
$IV$. $A'(t) < 0$ for all $t$.

Let $f(\alpha) = \int_{0}^{\alpha} x^{2} \left(1 - \frac{x}{\alpha}\right)^{\alpha} dx$ (where $\alpha > 0$),then $\sum_{\alpha=1}^{5} \frac{f(\alpha)}{\alpha^{3}}$ is equal to-

The number of continuous functions $f : [0, \frac{3}{2}] \rightarrow (0, \infty)$ satisfying the equation $4 \int_0^{3/2} f(x) dx + 125 \int_0^{3/2} \frac{dx}{\sqrt{f(x)+x^2}} = 108$ is

Let $u = \int_{0}^{\infty} \frac{dx}{x^4 + 7x^2 + 1}$ and $v = \int_{0}^{\infty} \frac{x^2 dx}{x^4 + 7x^2 + 1}$. Then:

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