1. ## integrate ln function

integrate ln(x^2-1)

2. Use the rules of exponents to rewrite as

$\displaystyle ln(x^{2}-1)=ln((x+1)(x-1))=ln(x+1)+ln(x-1)$

3. Originally Posted by galactus
Use the rules of exponents to rewrite as

$\displaystyle ln(x^{2}-1)=ln((x+1)(x-1))=ln(x+1)+ln(x-1)$
But care needs to be taken with domain issues since the original function is defined for x > 1 or x < -1 whereas the re-write has problems when x < -1 ....

4. Yep. I was being lazy and careless.

5. so how should I go about solving this problem?

6. Do exactly what galactus said but be careful like mr fantastic said!

$\displaystyle ln(x^2- 1)$ is an even function so you really only need to integrate for x> 1, where galactus' method has no problem, and then use the same integral for x< -1.

7. Or you can go with integration by parts

$\displaystyle \int{\ln(x^2-1)}$

sub. $\displaystyle \ln(x^2-1)=u \Rightarrow \frac{2x dx}{x^2-1}=du; dv = dx \Rightarrow v = x;$

so we have

$\displaystyle x\ln(x^2-1)-2\int{\frac{x^2}{x^2-1}dx}$
$\displaystyle = x\ln(x^2-1)-2\int{\frac{x^2-{\color{red}1}+{\color{red} 1}}{x^2-1}dx}$
$\displaystyle =x\ln(x^2-1)-2\int{\left (1+\frac{1}{x^2-1}\right )dx} =...$

correct me anyone if I did anything wrong.