Question

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

Answer 1

Consider

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

Now,

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

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