The average ordinate of $y = \sin x$ over $[0, \pi]$ is

$\int_0^{\pi / 4} \tan ^2(x) \, dx =$

Let $f(x) = \int\limits_0^x {{e^{ - {t^2}}}dt} $ for all $x > 0$. Then for all $x > 0$:

The integral $\int_{0}^{1} \frac{1}{7^{\left[\frac{1}{x}\right]}} dx$,where $[.]$ denotes the greatest integer function,is equal to:

The integral $\int_{1/4}^{3/4} \cos \left(2 \cot^{-1} \sqrt{\frac{1-x}{1+x}}\right) dx$ is equal to:

Let $[.]$ denote the greatest integer function. If $\int_0^{e^3}\left[\frac{1}{e^{x-1}}\right] d x=\alpha-\log _e 2$,then $\alpha^3$ is equal to . . . . . . .