The value of the integral $\int_{0}^{0.9} [x - 2[x]] \, dx$,where $[.]$ denotes the greatest integer function,is

All values of $a$ for which the inequality $\frac{1}{\sqrt{a}} \int_{1}^{a} (\frac{3}{2} \sqrt{x} + 1 - \frac{1}{\sqrt{x}}) dx < 4$ is satisfied, lie in the interval

$\int_0^{\pi /4} {\frac{{\sec x}}{{1 + 2{{\sin }^2}x}}} dx$ is equal to

$\int_0^{\pi / 4} \sqrt{1-\sin 2 x} \,d x =$

The value of the integral $\int_{1}^{3} [x^{2}-2x-2] dx$,where $[x]$ denotes the greatest integer function,is:

$\int_{0}^{3} \frac{3x + 1}{x^2 + 9} dx = $