By using the properties of definite integrals,evaluate the integral $\int_{0}^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1+\sin x \cos x} d x$.

The value of $\int\limits_0^1 {\sqrt[3]{{2{x^3} - 3{x^2} - x + 1}}\,dx} $ is

If $f(\alpha) = \int_{1}^{\alpha} \frac{\log_{10} t}{1+t} dt, \alpha > 0$,then $f(e^{3}) + f(e^{-3})$ is equal to.

Let $\int_{-2}^{2} (|\sin x| + [x \sin x]) dx = 2(3 - \cos 2) + \beta$,where $[\cdot]$ denotes the greatest integer function. Then $\beta \sin(\frac{\beta}{2})$ equals:

The value of $\int_{-2}^{2} \left[ p \ln \left( \frac{1+x}{1-x} \right) + q \ln \left( \frac{1-x}{1+x} \right)^{-2} + r \right] dx$ depends on:

If $f(t) = \int_0^\pi \frac{2x \, dx}{1 - \cos^2 t \sin^2 x}$,where $0 < t < \pi$,then the value of $\int_0^{\frac{\pi}{2}} \frac{\pi^2 \, dt}{f(t)}$ equals..........