Prove that $$\int_0^1 {{{\tan }^{ - 1}}} \,\left( {{1 \over {1 - x + {x^2}}}} \right)dx = 2\int_0^1 {{{\tan }^{ - 1}}} \,x\,dx.$$
Hence or otherwise, evaluate the integral
$$\int_0^1 {{{\tan }^{ - 1}}\left( {1 - x + {x^2}} \right)dx.} $$

Let $$f\left( x \right) = x - \left[ x \right],$$ for every real number $$x$$, where $$\left[ x \right]$$ is the integral part of $$x$$. Then $$\int_{ - 1}^1 {f\left( x \right)\,dx} $$ is

An n-digit number is a positive number with exactly digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is

Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent.