The value of the function $f(x) = 1 + x + \int\limits_1^x (\ln^2 t + 2 \ln t) \, dt$ where $f'(x)$ vanishes is:

Let a differentiable function $f$ satisfy the equation $\int_{0}^{36} f(\frac{tx}{36}) dt = 4\alpha f(x)$. If $y = f(x)$ is a standard parabola passing through the points $(2, 1)$ and $(-4, \beta)$, then $\beta^{\alpha}$ is equal to . . . . . . .

If $f(x) = \int_0^x {t\sin t\,dt} $,then $f'(x) = $

$\int_0^\pi x \sin^4 x \cos^6 x \, dx =$

Let $f:(0, \infty) \rightarrow \mathbb{R}$ be given by $f(x)=\int_{\frac{1}{x}}^x e^{-\left(t+\frac{1}{t}\right)} \frac{d t}{t}$. Then
$(A)$ $f(x)$ is monotonically increasing on $[1, \infty)$
$(B)$ $f(x)$ is monotonically decreasing on $(0,1)$
$(C)$ $f(x)+f\left(\frac{1}{x}\right)=0$,for all $x \in(0, \infty)$
$(D)$ $f\left(2^x\right)$ is an odd function of $x$ on $\mathbb{R}$

$\int_0^a x(2ax - x^2)^{3/2} dx = $