The value of $$\int_1^{{e^{37}}} {{{\pi \sin \left( {\pi In\,x} \right)} \over x}\,dx} $$ is ...............

For $$n>0,$$ $$\int_0^{2\pi } {{{x{{\sin }^{2n}}x} \over {{{\sin }^{2n}}x + {{\cos }^{2n}}x}}} dx = $$

If for nonzero $$x$$, $$af(x)+$$ $$bf\left( {{1 \over x}} \right) = {1 \over x} - 5$$ where $$a \ne b,$$ then
$$\int_1^2 {f\left( x \right)dx} = .......$$

The value of $$\int\limits_2^3 {{{\sqrt x } \over {\sqrt {3 - x} + \sqrt x }}} dx$$ is ...........