The value of $$\int_0^{\pi} \|cos x|^3 \; dx$$ is:

$$I = \int_{0}^{\pi} \vert{}\cos x\vert{}^3 \, dx$$

$$I = 2\int_{0}^{\pi/2} \vert{}\cos x\vert{}^3 \, dx$$

$$I = 2\int_{0}^{\pi/2} \cos^3 x \, dx$$

Using the identity $$\cos^3 x = \frac{\cos 3x + 3\cos x}{4}$$:

$$I = 2 \left[ \frac{\sin 3x}{12} + \frac{3\sin x}{4} \right]_{0}^{\pi/2}$$

$$I = 2 \left( \left( \frac{\sin(3\pi/2)}{12} + \frac{3\sin(\pi/2)}{4} \right) - (0) \right)$$

$$I = 2 \left( -\frac{1}{12} + \frac{3}{4} \right) = 2 \left( \frac{-1 + 9}{12} \right) = 2 \left( \frac{8}{12} \right) = \frac{4}{3}$$