$\displaystyle \int_{0}^{\frac{3\pi}{2}}cosx d(|cos2x|) $
pls can anybody help me with a book link or material that treat this sort of problem above, am having challenges with it.
thanks.
there's no particular trick to this. You just need to observe where within the limits of integration |cos2x| changes sign and account for it.
$\displaystyle \cos 2x \geq 0$ on [0,pi/4], [3pi/4, 5pi/4]
$\displaystyle \cos 2x \leq 0$ on [pi/4, 3pi/4], [5pi/4, 6pi/4]
so split the integral up into these pieces using $\displaystyle \pm \cos 2x$ as appropriate.
This is a nasty problem. Look at this webpage.
On your question, I would do the integration by parts formula first.
Then use the derivative formula (just above that) on the new integral.