Let P(x) be a real polynomial of degree 3 which vanishes at x = $$-$$3. Let P(x) have local minima at x = 1, local maxima at x = $$-$$1 and $$\int\limits_{ - 1}^1 {P(x)dx} $$ = 18, then the sum of all the coefficients of the polynomial P(x) is equal to _________.

Let f(x) and g(x) be two functions satisfying f(x²) + g(4 $$-$$ x) = 4x³ and g(4 $$-$$ x) + g(x) = 0, then the value of $$\int\limits_{ - 4}^4 {f{{(x)}^2}dx} $$ is

Let $${I_n} = \int_1^e {{x^{19}}{{(\log |x|)}^n}} dx$$, where n$$\in$$N. If (20)I₁₀ = $$\alpha$$I₉ + $$\beta$$I₈, for natural numbers $$\alpha$$ and $$\beta$$, then $$\alpha$$ $$-$$ $$\beta$$ equals to ___________.

If [ . ] represents the greatest integer function, then the value of


$$\left| {\int\limits_0^{\sqrt {{\pi \over 2}} } {\left[ {[{x^2}] - \cos x} \right]dx} } \right|$$ is ____________.