# Thread: How to calculate int(ln(x)/(x+1))?

1. ## How to calculate int(ln(x)/(x+1))?

Does anybody know how to calculate the integral $\displaystyle \int{\frac{lnx}{1+x}}$? Thanks.

2. Originally Posted by curvature
Does anybody know how to calculate the integral $\displaystyle \int{\frac{lnx}{1+x}}$? Thanks.
I don't think it can be expressed using a finite number of elementary functions. Perhaps it's meant to be a definite integral ....?

3. Originally Posted by curvature
Does anybody know how to calculate the integral $\displaystyle \int{\frac{lnx}{1+x}}$? Thanks.
According to Mathematica,

$\displaystyle \int{\frac{\ln{x}}{1 + x}\,dx} = \ln{x}\ln{(1 + x)} + \textrm{Li}_2(-x)$

where $\displaystyle \textrm{Li}_2(-x) = \sum_{k = 1}^{\infty}{\frac{(-x)^k}{k^2}}$

This is the Polylogarithm function, see Polylogarithm - Wikipedia, the free encyclopedia.

I do not know how it is calculated to be this function though...

4. Originally Posted by mr fantastic
I don't think it can be expressed using a finite number of elementary functions. Perhaps it's meant to be a definite integral ....?
I mean an indefinite integral.

5. Originally Posted by curvature
I mean an indefinite integral.
Are you looking for an answer like the one in post #3?

6. Originally Posted by mr fantastic
Are you looking for an answer like the one in post #3?
Perhaps post#3 offers the best answer since I no concise form is possible to the integral.

7. Originally Posted by curvature
Perhaps post#3 offers the best answer since I no concise form is possible to the integral.
Are you sure you copied down the integral correctly?

$\displaystyle \int{\frac{\ln{x}}{x + 1}\,dx}$ is hard, but

$\displaystyle \int{\frac{\ln{x}}{x}\,dx}$ is easy, you just use a u-substitution...

8. Originally Posted by Prove It
Are you sure you copied down the integral correctly?

$\displaystyle \int{\frac{\ln{x}}{x + 1}\,dx}$ is hard, but

$\displaystyle \int{\frac{\ln{x}}{x}\,dx}$ is easy, you just use a u-substitution...
I mean the first one. Thank you.