$$\int\limits_0^\pi {{{\left| {\cos x} \right|}^3}} \,dx$$

The period of $$\left| {\cos x} \right|$$ = $${\pi \over 2}$$

$$ \therefore $$  I = 2 $$\int\limits_0^{{\pi \over 2}} {{{\left| {\cos x} \right|}^3}} \,dx$$

as in the range 0 to $${\pi \over 2}$$ $$\left| {\cos x} \right|$$ is positive.

So,     $$\left| {\cos x} \right|$$ = $$cosx$$

$$ \therefore $$  I = 2 $$\int\limits_0^{{\pi \over 2}} {{{\cos }^3}x\,dx} $$

= 2$$\int\limits_0^{{\pi \over 2}} {\left( {{{\cos 3x + 3\cos x} \over 4}} \right)} dx$$

I = $${1 \over 2}\left[ {{{\sin 3x} \over 3} + 3\sin x} \right]_0^{{\pi \over 2}}$$

I = $${1 \over 2}\left[ {{1 \over 3}\left( {{{3\pi } \over 2}} \right) + 3.\sin {\pi \over 2}} \right]$$

I = $${1 \over 2}\left[ { - {1 \over 3} + 3} \right]$$

= $${1 \over 2}\left( {{8 \over 3}} \right)$$

= $${4 \over 3}$$