Question

${\int }_0^{\mathrm{\pi }}\cos x\left|\cos x\right|dx$

Answer 1

Consider $f\left(x\right)=\cos x\left|\cos x\right|$

Now,

$f\left(\mathrm{\pi }-x\right)=\cos \left(\mathrm{\pi }-x\right)\left|\cos \left(\mathrm{\pi }-x\right)\right|=-\cos x\left|-\cos x\right|=-\cos x\left|\cos x\right|=-f\left(x\right)$

$\therefore {\int }_0^{\mathrm{\pi }}\cos x\left|\cos x\right|dx=0\left[{\int }_0^{2a}f\left(x\right)dx=\left\{\begin{array}{ll}2{\int }_0^{a}f\left(x\right)dx,& \mathrm{if}f\left(2a-x\right)=f\left(x\right)\\ 0,& \mathrm{if}f\left(2a-x\right)=-f\left(x\right)\end{array}\right.\right]$