The value of the definite integral $\int_{1}^{\infty} (e^{x+1} + e^{3-x})^{-1} \, dx$ is

Let $f:[0,1] \rightarrow [0,1]$ be a continuous function such that $x^2+(f(x))^2 \leq 1$ for all $x \in [0,1]$ and $\int_0^1 f(x) dx = \frac{\pi}{4}$. Then,$\int_{\frac{1}{2}}^{\frac{1}{\sqrt{2}}} \frac{f(x)}{1-x^2} dx$ equals

Number of ordered pairs $(a, b)$ satisfying simultaneously the system of equations $\int_{a}^{b} x^{3} dx = 0$ and $\int_{a}^{b} x^{2} dx = \frac{2}{3}$ is

If $\int_{0}^{k} \frac{dx}{2 + 18x^2} = \frac{\pi}{24}$,then the value of $k$ is

The value of the definite integral $\int \limits_0^{\pi / 2} \frac{\sin x \cos x}{1+\cos ^4 x} d x$ is:

The velocity of a particle which starts from rest is given by the following table. The total distance travelled (in metre) by the particle in $10 \ s$, using the Trapezoidal rule, is:

$t \ (\text{in second})$	$0$	$2$	$4$	$6$	$8$	$10$
$v \ (\text{in m/s})$	$0$	$12$	$16$	$20$	$35$	$60$