Determine if the following integral is convergent or divergent. If it is convergent find its value.

Question

The next week I have a test over Improper Integrals and yesterday started to solve some problems that I've found on th Internet. Everything was fine until I stuck on this example:

Determine if the following integral is convergent or divergent. If it is convergent find its value - $$\int^{\pi/2}_{0} \ln(\cos x)dx$$

Any ideas how to test the integral is convergent or divergent?

Compare it to $\int_0^{\pi/2}\log\sin x,dx$, and compare that one to $\int_0^{\pi/2}\log x,dx$. — Gerry Myerson, Aug 31 '13 at 12:51
@Gerry Myerson - sorry but I didn't get it. Could explain a bit more? — NoSense, Aug 31 '13 at 12:55
You thought about my suggestions for all of 4 minutes. Please, think about them seriously. — Gerry Myerson, Aug 31 '13 at 13:02
Sorry, but I can't understand why I have to compare it with logsinx — NoSense, Aug 31 '13 at 13:12
@NoSense You don't have to compare it to that. Rather, you could transform it to that ($\theta = \pi/2 - x$), and then comparing $\sin \theta$ to $\theta$ shows the convergence. — Daniel Fischer, Aug 31 '13 at 14:09

score 0 · Accepted Answer · answered Aug 31 '13 at 13:38

0

Verify that, on $[0,\pi/2]$

$$\tag 1 0\geq \log\cos x\geq \log\left(1-\frac 2\pi x\right)$$

and the latter is an integrable lower bound. The function $$\varphi(\varepsilon)=\int_0^{\pi/2-\varepsilon}\log\cos x dx$$ decreases monotonically as $\varepsilon \to\frac \pi 2^-$, and is bounded below by $(1)$, so it converges.

answered Aug 31 '13 at 13:38

Pedro

122,002

How do I know that logcosx > log(1-2/pi*x)? – NoSense Aug 31 '13 at 13:45
@NoSense It suffices that you check that $\cos x\geq 1-\frac 2\pi x$. Can you do that? – Pedro Aug 31 '13 at 15:17
aha, I will try to do it myself, thanks for the help – NoSense Aug 31 '13 at 16:35

Determine if the following integral is convergent or divergent. If it is convergent find its value.

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