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:

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 $2$.

The value of $\int\limits_0^2 {\frac{{dx}}{{{{(1 - x)}^2}}}} $ is

If $A_n = \int_{0}^{\pi /2} \frac{\sin((2n-1)x)}{\sin x} dx$ and $B_n = \int_{0}^{\pi /2} \left( \frac{\sin(nx)}{\sin x} \right)^2 dx$ for $n \in N$,then:

Let $\operatorname{Max} \limits _{0 \leq x \leq 2}\left\{\frac{9-x^{2}}{5-x}\right\}=\alpha$ and $\operatorname{Min} \limits _ {0 \leq x \leq 2}\left\{\frac{9-x^{2}}{5-x}\right\}=\beta$. If $\int\limits_{\beta-\frac{8}{3}}^{2 \alpha-1} \operatorname{Max}\left\{\frac{9- x ^{2}}{5- x }, x \right\} dx =\alpha_{1}+\alpha_{2} \log _{e}\left(\frac{8}{15}\right)$,then $\alpha_{1}+\alpha_{2}$ is equal to

Let $f: R \rightarrow R$ be defined as $f(x)=a e^{2 x}+b e^x+c x$. If $f(0)=-1$,$f^{\prime}(\log _e 2)=21$,and $\int_0^{\log _e 4}(f(x)-c x) d x=\frac{39}{2}$,then the value of $|a+b+c|$ equals: