For any integer $$n$$ the integral ...........
$$\int\limits_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}\left( {2n + 1} \right)xdx} $$ has the value

The value of the integral $$\int\limits_0^{\pi /2} {{{\sqrt {\cot x} } \over {\sqrt {\cot x} + \sqrt {\tan x} }}dx} $$ is

The area bounded by the curves $$y=f(x)$$, the $$x$$-axis and the ordinates $$x=1$$ and $$x=b$$ is $$(b-1)$$ sin $$(3b+4)$$. Then $$f(x)$$ is

Let $$a, b, c$$ be non-zero real numbers such that
$$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} } $$
Then the quadratic equation $$a{x^2} + bx + c = 0$$ has