If $\int \limits_{\frac{1}{3}}^3 |\log_e x| dx = \frac{m}{n} \log_e \left(\frac{n^2}{e}\right)$,where $m$ and $n$ are coprime natural numbers,then $m^2 + n^2 - 5$ is equal to $............$.

$\int_1^2 \left( \tan ^{-1}\left(\frac{x}{x^2+1}\right)+\tan ^{-1}\left(\frac{x^2+1}{x}\right) \right) d x =$

Evaluate the definite integral $\int_{0}^{\frac{\pi}{4}} \frac{\sin x+\cos x}{9+16 \sin 2 x} d x$.

If $24 \int_0^{\frac{\pi}{4}} \left( \sin \left| 4x - \frac{\pi}{12} \right| + [2 \sin x] \right) dx = 2 \pi + \alpha$,where $[\cdot]$ denotes the greatest integer function,then $\alpha$ is equal to . . . . . .

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:

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

Suppose that $F(x)$ is an antiderivative of $f(x) = \frac{\sin x}{x}$,$x > 0$. Then $\int_{1}^{3} \frac{\sin 2x}{x} dx$ can be expressed as: