The value of $\int \limits_{0}^{\pi}|\cos x|^{3} dx$ is

$\mathop {Lim}\limits_{k \to 0} \frac{1}{k} \int\limits_0^k (1 + \sin 2x)^{\frac{1}{x}} dx$

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

Let $F(x) = \int_x^{x^2+\frac{\pi}{6}} 2 \cos^2 t \, dt$ for all $x \in \mathbb{R}$ and $f: [0, \frac{1}{2}] \rightarrow [0, \infty)$ be a continuous function. For $a \in [0, \frac{1}{2}]$,if $F'(a) + 2$ is the area of the region bounded by $x=0, y=0, y=f(x)$ and $x=a$,then $f(0)$ is

The points of extremum of $\int_0^{x^2} \frac{t^2 - 5t + 4}{2 + e^t} \,dt$ are