(i) Substitute \(u = \sin x\) and \(du = \cos x \, dx\). The integrand becomes \(e^{2u}\).

The indefinite integral is \(\frac{1}{2} e^{2u}\).

Using limits \(u = 0\) and \(u = 1\), the definite integral is \(\frac{1}{2}(e^2 - 1)\).

(ii) Use the chain rule or product rule to find the derivative: \(2\cos x \, e^{2\sin x} \cos x - e^{2\sin x} \sin x\).

Equate the derivative to zero and solve the quadratic equation in \(\sin x\).

Solve the 3-term quadratic to find \(x = 0.896\).