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  • I need help with - The value ofis : - Integral Calculus - JEE Main

 

The value of \int_{0}^{\pi}|\cos x|^3dx is :

  • Option 1)

     

    0

  • Option 2)

    \frac{4}{3}

  • Option 3)

    \frac{2}{3}

  • Option 4)

    -\frac{4}{3}

Answers (1)

best_answer

 

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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