# The value of $\int_{0}^{\pi}|\cos x|^3dx$ is :Option 1)  0Option 2)$\frac{4}{3}$Option 3)$\frac{2}{3}$Option 4)$-\frac{4}{3}$

Fundamental Properties of Definite integration -

If the function is continuous in (a, b ) then integration of a function a to b will be same as the sum of integrals of the same function from a to c and c to b.

$\int_{b}^{a}f\left ( x \right )dx= \int_{a}^{c}f\left ( x \right )dx+\int_{c}^{b}f\left ( x \right )dx$

- wherein

$Y=\cos \left ( x \right )$

$\int_{0}^{\pi }\left | \cos \left ( x \right ) \right |^{3}dx=\int_{0}^{\frac{\pi }{2}}\cos ^{3}\left ( x \right )dx-\int_{\frac{\pi }{2}}^{\pi }\cos ^{3}xdx$

$\because \cos \left ( 3x \right )=4\cos ^{3}\left ( x \right )-3\cos \left ( x \right )$

$=\int_{0}^{\frac{\pi }{2}}\left ( \frac{\cos \left ( 3x \right )+3\cos \left ( x \right )}{4} \right )dx-\int_{\frac{\pi }{2}}^{\pi }\left ( \frac{\cos \left ( 3x \right )+3\cos \left ( x \right )}{4} \right )dx$

$=\frac{1}{4}\left [ \left ( \frac{\sin \left ( 3x \right )}{3}+3\sin \left ( x \right ) \right )_{0}^{\frac{\pi }{2}} -\left ( \frac{\sin \left ( 3x \right )}{3}+3\sin \left ( x \right ) \right )_{\frac{ \pi }{2}}^{\pi }\right ]$

$=\frac{4}{3}$

Option 1)

0

Option 2)

$\frac{4}{3}$

Option 3)

$\frac{2}{3}$

Option 4)

$-\frac{4}{3}$

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