Let $${I_n} = \int\limits_0^1 {{x^n}} {\tan ^{ - 1}}xdx$$. If $${a_n}{I_{n + 2}} + {b_n}{I_n} = {c_n}$$ for all n $$ \ge $$ 1, then

Let $$I = \int\limits_0^I {{{{x^3}\cos 3x} \over {2 + {x^2}}}dx} $$, then

Let f be a non-constant continuous function for all x $$ \ge $$ 0. Let f satisfy the relation f(x) f(a $$-$$ x) = 1 for some a $$ \in $$ R⁺. Then, $$I = \int_0^a {{{dx} \over {1 + f(x)}}} $$ is equal to

If $$\phi (t) = \left\{ \matrix{ 1,\,for\,0 \le t < 1, \hfill \cr 0,\,otherwise \hfill \cr} \right.$$, then $$\int\limits_{ - 300}^{3000} {\left( {\sum\limits_{r' = 2014}^{2016} {\phi (t - r')\phi (t - 2016)} } \right)} \,dt$$ is