The number of solutions of the equation $6 \int_{0}^{|x|} ((t^2-1) \ln t) dt = 5|x|$ for $x \in R \setminus \{0\}$ is

If $\int_{0}^{\pi} (\sin^{3} x) e^{-\sin^{2} x} dx = \alpha - \frac{\beta}{e} \int_{0}^{1} \sqrt{t} e^{t} dt$,then $\alpha + \beta$ is equal to $....$

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

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 $2$.

Which of the following inequalities is/are $TRUE$?
$(A)$ $\int_0^1 x \cos x \, dx \geq \frac{3}{8}$
$(B)$ $\int_0^1 x \sin x \, dx \geq \frac{3}{10}$
$(C)$ $\int_0^1 x^2 \cos x \, dx \geq \frac{1}{2}$
$(D)$ $\int_0^1 x^2 \sin x \, dx \geq \frac{2}{9}$