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Question

\[\int_0^\pi \cos x\left| \cos x \right|dx\]

Sum

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Solution

Consider

\[f\left( x \right) = \cos x\left| \cos x \right|\]

Now,

\[f\left( \pi - x \right) = \cos\left( \pi - x \right)\left| \cos\left( \pi - x \right) \right| = - \cos x\left| - \cos x \right| = - \cos x\left| \cos x \right| = - f\left( x \right)\]

\[\therefore \int_0^\pi \cos x\left| \cos x \right|dx = 0 ..................\left[ \int_0^{2a} f\left( x \right)dx = \begin{cases}2 \int_0^a f\left( x \right)dx, & \text{if }f\left( 2a - x \right) = f\left( x \right) \\ 0, & \text{if }f\left( 2a - x \right) = - f\left( x \right)\end{cases} \right]\]

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Definite Integrals

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Chapter 20: Definite Integrals - Exercise 20.3 [Page 56]

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RD Sharma Mathematics [English] Class 12

Chapter 20 Definite Integrals
Exercise 20.3 | Q 23 | Page 56

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