(a) To find the \(x\)-coordinate of \(M\), we first differentiate \(y = \sin x \cos 2x\) using the product rule:

\(\frac{dy}{dx} = \cos x \cdot \cos 2x - \sin x \cdot 2\sin 2x\).

Set the derivative to zero to find critical points:

\(\cos x \cdot \cos 2x - \sin x \cdot 2\sin 2x = 0\).

Using the double angle formula \(\cos 2x = 2\cos^2 x - 1\), we simplify to:

\(6\sin^2 x = 1\) or \(6\cos^2 x = 5\) or \(5\tan^2 x = 1\).

Solving gives \(x = 0.421\) to 3 significant figures.

(b) Using the substitution \(u = \cos x\), we have \(du = -\sin x \, dx\).

The integral becomes \(\int (2u^2 - 1) \, du\).

Integrating, we get \(\frac{u}{3} - \frac{2}{3}u^3\).

Using limits \(u = 1\) and \(u = \frac{1}{\sqrt{2}}\), we evaluate:

\(\frac{1}{3}(\sqrt{2} - 1)\).