Let $$ABC$$ be a triangle having $$O$$ and $$I$$ as its circumcenter and in centre respectively. If $$R$$ and $$r$$ are the circumradius and the inradius, respectively, then prove that $${\left( {IO} \right)^2} = {R^2} - 2{\mathop{\rm Rr}\nolimits} $$. Further show that the triangle BIO is a right-angled triangle if and only if $$b$$ is arithmetic mean of $$a$$ and $$c$$.

The function $$f(x)=$$ $${\sin ^4}x + {\cos ^4}x$$ increases if

Integrate $$\int {{{{x^3} + 3x + 2} \over {{{\left( {{x^2} + 1} \right)}^2}\left( {x + 1} \right)}}dx.} $$

$$\int\limits_{\pi /4}^{3\pi /4} {{{dx} \over {1 + \cos x}}} $$ is equal to