If \( f(x) = \int \frac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2} \, dx , \; (x \geq 0), \; f(0) = 0 \) and \( f(1) = \frac{1}{k} \), then the value of \( K \) is ____________.
If \( \int \dfrac{dx}{(x^2 + x + 1)^2} = a \tan^{-1}\!\left( \dfrac{2x + 1}{\sqrt{3}} \right) + b \left( \dfrac{2x + 1}{x^2 + x + 1} \right) + C,\ x > 0 \) where \( C \) is the constant of integration, then the value of \( 9(\sqrt{3}\,a + b) \) is equal to __________.
