$\int_1^2 \frac{1}{x^2} e^{-\frac{1}{x}} \, dx = $

The value of $\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \frac{{\int_{\pi /2}^x {t\,dt} }}{{\sin (2x - \pi )}}$ is

Choose the correct option regarding the following definite integrals:
$(i)$ $\int_0^{\pi / 2} \sin ^m(x) \cos (x) d x = \frac{1}{m+1}$
(ii) $\int_0^{\pi / 2} \sin (x) \cos ^n(x) d x = \frac{1}{n+1}$

If $f$ is integrable on $[0, a]$,then the function $h$ defined on $[0, a]$ as $h(x) = \int_0^x f(t) dt$ is integrable on $[0, a]$. Which of the following functions is also integrable on $[0, a]$?

The set of values of $a$ which satisfy the equation $\int_{0}^{2} (t - \log_{2} a) \, dt = \log_{2} \left( \frac{4}{a^{2}} \right)$ is

$\int_{-1}^{1} |1 - x| \,dx = $