Let $f(x)=x+\frac{a}{\pi^2-4} \sin x+\frac{b}{\pi^2-4} \cos x$ for $x \in R$ be a function which satisfies $f(x)=x+\int \limits_0^{\pi / 2} \sin (x+y) f(y) d y$. Then $(a+b)$ 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 $2$.

Let $u = \int_0^1 \frac{\ln(x + 1)}{x^2 + 1} \, dx$ and $v = \int_0^{\frac{\pi}{2}} \ln(\sin 2x) \, dx$,then:

If $f(x)$ satisfies the relation $f(x) = e^{x} + \int_{0}^{1} (y + xe^{x}) f(y) dy$, then $e + f(0)$ is equal to . . . . . . .

If $f(x) = \int_0^x {t(\sin x - \sin t) dt}$,then which of the following is true?

If $x$ satisfies the equation $\left( \int_{0}^{1} \frac{dt}{t^2 + 2t \cos \alpha + 1} \right) x^2 - \left( \int_{-3}^{3} \frac{t^2 \sin 2t}{t^2 + 1} dt \right) x - 2 = 0$ for $0 < \alpha < \pi$,then the value of $x$ is