How to Approach Integrals with Absolute Values in Limits?

[dirk_mec1]

Homework Statement

Evaluate:

<br /> <br /> \lim_{n\to\infty} \int_0^{\frac{\pi}{2}} x|\cos nx|\ \mathrm{d}x<br /> <br />

Homework Equations

hint: the integral is not zero.

The Attempt at a Solution

I don't know how to start: how do I deal with the absolute sign?

 

[tnutty]

make two cases. One for the positive and the other for the negative

 

[dirk_mec1]

How do I know where it becomes negative?

 

[tnutty]

1) all positive terms

2) bring out the negative outside of the integral.

 

[Dick]

How do I know where it becomes negative?

It changes sign everywhere cos(nx) vanishes, when nx is an odd multiple of pi/2. You might find it easier to count if you do the change of variables u=nx first. Then follow tnutty's advice and add up the positive parts and negative parts separately. Try and guess the answer first. For large n you get many cosine cycles. So it ought to be the integral from 0 to pi/2 of x*(the average value of |cos|).