A) \[-\frac{4}{3}\]
B) 0
C) \[\frac{4}{3}\]
D) \[\frac{2}{3}\]
Correct Answer: C
Solution :
\[I\,\,=\,\,\int\limits_{0}^{\pi }{{{\left| \cos x \right|}^{3}}}dx\] \[I\,\,=\,\,2\,\,\int\limits_{0}^{\pi /2}{{{\left| \cos x \right|}^{3}}}dx\] \[I\,\,=\,\,2\,\,\int\limits_{0}^{\pi /2}{{{\cos }^{3\,}}x\,.\,}dx\] \[I\,\,=\,\,2\,\,\int\limits_{0}^{\pi /2}{\left( \frac{\cos \,3\,x\,\,+\,\,\cos \,\,x}{4} \right)\,\,.\,\,dx}\] \[I\,\,=\,\frac{1}{2}\,{{\left[ \frac{\sin \,3x}{3}\,+\,3\,\,\sin \,\,x \right]}_{\,\,0}}^{\pi /2}\] \[I\,\,=\,\,\frac{1}{2}\,\left[ \frac{1}{3}\,.\,\sin \,3.\,\frac{\pi }{2}\,\,+\,\,3.1 \right]\] \[I\,\,=\,\,\frac{1}{2}\,\left[ \frac{-1}{3}\,\,+3 \right]\] \[=\,\,\frac{1}{2}\,\left( \frac{8}{3} \right)\,\,=\,\,\frac{4}{3}\]
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