Question
The value of $$\int\limits_0^\pi {{{\left| {\cos \,x} \right|}^3}dx}, $$ is :
A.
$$0$$
B.
$$\frac{4}{3}$$
C.
$$\frac{2}{3}$$
D.
$$ - \frac{4}{3}$$
Answer :
$$\frac{4}{3}$$
Solution :
$$\eqalign{
& I = \int\limits_0^\pi {{{\left| {\cos \,x} \right|}^3}dx} = 2\int\limits_0^{\frac{\pi }{2}} {{{\cos }^3}\,x\,dx} \cr
& = \frac{2}{4}\int\limits_0^{\frac{\pi }{2}} {\left( {3\,\cos \,x + \cos \,3x} \right)dx} \cr
& \left[ {\lambda \,\cos \,3\theta = 4\,{{\cos }^3}\theta - 3\,\cos \,\theta } \right] \cr
& = \frac{1}{2}\left[ {3\,\sin \,x + \frac{{\sin \,3x}}{3}} \right]_0^{\frac{\pi }{2}} \cr
& = \frac{1}{2}\left( {3 - \frac{1}{3}} \right) \cr
& = \frac{4}{3} \cr} $$