If $f(x) = \text{Max}\{\sin x, \cos x\}$ and $g(x) = \text{Min}\{\sin x, \cos x\}$,then $\int_{0}^{\pi} f(x) dx + \int_{0}^{\pi} g(x) dx = $

If for a real number $y$,$[y]$ is the greatest integer less than or equal to $y$,then the value of the integral $\int_{\pi /2}^{3\pi /2} [2\sin x] \, dx$ is

If $A_n = \int_{\frac{\pi}{2}}^{\infty} e^{-x} \cos^n x \, dx$,then $\frac{A_4 - A_6}{A_4} = $

Let $F: R \rightarrow R$ be a thrice differentiable function. Suppose that $F(1)=0, F(3)=-4$ and $F'(x) < 0$ for all $x \in (1/2, 3)$. Let $f(x)=x F(x)$ for all $x \in R$.
$1.$ The correct statement$(s)$ is(are):
$(A) f'(1) < 0$
$(B) f(2) < 0$
$(C) f'(x) \neq 0$ for any $x \in (1, 3)$
$(D) f'(x)=0$ for some $x \in (1, 3)$
$2.$ If $\int_1^3 x^2 F '(x) dx = -12$ and $\int_1^3 x^3 F''(x) dx = 40$,then the correct expression$(s)$ is(are):
$(A) 9 f'(3)+f'(1)-32=0$
$(B) \int_1^3 f(x) dx = 12$
$(C) 9 f'(3)-f'(1)+32=0$
$(D) \int_1^3 f(x) dx = -12$
Give the answer for question $1$ and $2$.

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-

Let $f(x)=(1-x)^2 \sin ^2 x+x^2$ for all $x \in \mathbb{R}$ and let $g(x)=\int_1^x \left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) dt$ for all $x \in (1, \infty)$.
$1.$ Which of the following is true?
$(A)$ $g$ is increasing on $(1, \infty)$
$(B)$ $g$ is decreasing on $(1, \infty)$
$(C)$ $g$ is increasing on $(1,2)$ and decreasing on $(2, \infty)$
$(D)$ $g$ is decreasing on $(1,2)$ and increasing on $(2, \infty)$
$2.$ Consider the statements:
$P$ : There exists some $x \in \mathbb{R}$ such that $f(x)+2x=2(1+x^2)$
$Q$ : There exists some $x \in \mathbb{R}$ such that $2f(x)+1=2x(1+x)$
Then
$(A)$ both $P$ and $Q$ are true
$(B)$ $P$ is true and $Q$ is false
$(C)$ $P$ is false and $Q$ is true
$(D)$ both $P$ and $Q$ are false
Give the answer for question $1$ and $2$.