Show that $$\int\limits_0^\pi {xf\left( {\sin x} \right)dx} = {\pi \over 2}\int\limits_0^\pi {f\left( {\sin x} \right)dx.} $$

Find the value of $$\int\limits_{ - 1}^{3/2} {\left| {x\sin \,\pi \,x} \right|\,dx} $$

For any real $$t,\,x = {{{e^t} + {e^{ - t}}} \over 2},\,\,y = {{{e^t} - {e^{ - t}}} \over 2}$$ is a point on the
hyperbola $${x^2} - {y^2} = 1$$. Show that the area bounded by this hyperbola and the lines joining its centre to the points corresponding to $${t_1}$$ and $$-{t_1}$$ is $${t_1}$$.

If $$A$$ and $$B$$ are two events such that $$P\left( A \right) > 0,$$ and $$P\left( B \right) \ne 1,$$ then $$P\left( {{{\overline A } \over {\overline B }}} \right)$$ is equal to