A curve has equation

\(y=x\mathrm e^{2x}\).

(a) Find \(\frac{\mathrm dy}{\mathrm dx}\).

(b) Find the equation of the normal to the curve at the point where \(x=1\).

(c) Use your answer to part (a) to find the exact value of

\(\displaystyle \int_0^2 2x\mathrm e^{2x}\,\mathrm dx.\)