The ionisation constant of $$NH_4^+$$ in water is 5.6 $$\times$$ 10^-10 at 25^oC. The rate constant for the reaction of $$NH_4^+$$ and $$OH^-$$ to form NH₃ and H₂O at 25^oC is 3.4 $$\times$$ 10¹⁰ L mol^-1s^-1. Calculate the rate constant for proton transfer from water to NH₃.

For $$n>0,$$ $$\int_0^{2\pi } {{{x{{\sin }^{2n}}x} \over {{{\sin }^{2n}}x + {{\cos }^{2n}}x}}} dx = $$

Let $${A_n}$$ be the area bounded by the curve $$y = {\left( {\tan x} \right)^n}$$ and the
lines $$x=0,$$ $$y=0,$$ and $$x = {\pi \over 4}.$$ Prove that for $$n > 2,$$
$${A_n} + {A_{n - 2}} = {1 \over {n - 1}}$$ and deduce $${1 \over {2n + 2}} < {A_n} < {1 \over {2n - 2}}.$$

Determine the equation of the curve passing through the origin, in the form $$y=f(x),$$ which satisfies the differential equation $${{dy} \over {dx}} = \sin \left( {10x + 6y} \right).\,$$