# Help me with this logarithmic integral

Gold Member

## Homework Statement

$\displaystyle \int_0^{2 \pi} x \ln \dfrac{3+ \cos x}{3- \cos x} dx$

## The Attempt at a Solution

Using property of definite integral
2I = $\displaystyle \int_0^{2 \pi} 2 \pi \ln \dfrac{3+ \cos x}{3- \cos x} dx$

Homework Helper

## Homework Statement

$\displaystyle \int_0^{2 \pi} x \ln \dfrac{3+ \cos x}{3- \cos x} dx$

## The Attempt at a Solution

Using property of definite integral
2I = $\displaystyle \int_0^{2 \pi} 2 \pi \ln \dfrac{3+ \cos x}{3- \cos x} dx$

I'm not sure what "property of definite integral" would give you that. You'll have to spell it out. Here's a hint. Change the variable to u=2pi-x and see what happens.

Staff Emeritus
Homework Helper
You can also rewrite the log of a quotient into something simpler.

Gold Member
I'm not sure what "property of definite integral" would give you that. You'll have to spell it out. Here's a hint. Change the variable to u=2pi-x and see what happens.

$\int_0^a f(x) = \int_0^a f(a-x)$. This is what I've used.

Then I added both integrals to get rid of x outside log.

Gold Member
You can also rewrite the log of a quotient into something simpler.

Can you please elaborate? I didn't get you.

Hertz
$\int_0^a f(x) = \int_0^a f(a-x)$. This is what I've used.

Then I added both integrals to get rid of x outside log.

Integrating from one point to another finds the area under the curve between those two points. Think about the graphs of f(x) and f(a-x). f(a-x) is going to be flipped over the x-axis and shifted to the left by "a" units.. Most of the time, the value of the integral wouldn't be the same for both functions. Sorry to tell ya, but I think the property of definite integral may only apply to periodic functions. I don't think this function is periodic

Can you please elaborate? I didn't get you.

$log(\frac{a}{b})=log(a)-log(b)$
$log(a*b)=log(a)+log(b)$

Saitama
Using property of definite integral
2I = $\displaystyle \int_0^{2 \pi} 2 \pi \ln \dfrac{3+ \cos x}{3- \cos x} dx$

Okay. Now observe that
$$\int_0^{2 \pi} \ln \dfrac{3+ \cos x}{3- \cos x} dx=2\int_0^{\pi} \ln \dfrac{3+ \cos x}{3- \cos x} dx$$

Use the same property you used before.

Gold Member
Okay. Now observe that
$$\int_0^{2 \pi} \ln \dfrac{3+ \cos x}{3- \cos x} dx=2\int_0^{\pi} \ln \dfrac{3+ \cos x}{3- \cos x} dx$$

Use the same property you used before.

Well done.