Let $H(x) = \int_{x^2}^{x^3} (x + 1) \sin(t^3) dt$. Then $\lim_{x \to 1} \frac{H(x)}{x - 1}$ is equal to:

$\lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}}(\sin \sqrt{t}) dt }{x^{3}}$ is equal to

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 $\mathbb{R}$

If $I=\int_{-a}^a(x^4-2x^2)dx$,then $I$ is minimum at $a=$

$\int_0^\pi x \cdot \sin^5 x \cdot \cos^6 x \, dx =$

The value of the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^4 x \left( 1 + \log \left( \frac{2 + \sin x}{2 - \sin x} \right) \right) dx$ is