If $\int_0^1\left(\sum_{r=1}^{2013} \frac{x}{x^2+r^2}\right)\left(\prod_{r=1}^{2013}\left(x^2+r^2\right)\right) d x=\frac{1}{2}\left[\left(\prod_{r=1}^{2013}\left(1+r^2\right)-K^2\right]\right.$, then $K$ is

The least positive value of ' $a$ ' for which the equation $\int_0^x\left(t^2-8 t+13\right) d t=x \sin \frac{a}{x}$ has a solution is

Let $f(x)$ be a real valued $f$ unction which is monotonic and differentiable. Then for any reals a and $b, \int_{f(a)}^{f(b)} 2 x\left\{b-f^{-1}(x)\right\} d x=$

Let $a, b, c$ be non-zero real numbers, such that $\int_0^r\left(1+\cos ^8 x\right)\left(a x^2+b x+c\right) d x=\int_0^{2^{\prime}}\left(1+\cos ^8 x\right)\left(a x^2+b x+c\right) d x$, then $a x^2+b x+c=0$ has