- #1

Mayhem

- 322

- 213

- Homework Statement
- Evaluate ##\int{\ln{(e^x+1)}}##

- Relevant Equations
- ##\ln{x} = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}x^n}{n}##

##(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}y^k##

Given the integral $$\int \ln{(e^x+1)} dx$$ we can rewrite this as the integral of the Taylor expansion of ##\ln{(e^x+1)}##. $$\int \sum_{n=1}^{\infty} \frac{(-1)^{n-1}(e^x+1)^n}{n} dx$$ Which can then be rewritten using the binomial theorem: $$\int \sum_{n=1}^{\infty} \left [ \frac{(-1)^{n-1}}{n} \sum_{k=0}^{n} \binom{n}{k} e^{x(n-k)} \right] dx $$ What I want: a way to rewrite this such that I can directly integrate the binomial term, as this is simply a linear combination of ##e^{x(n-k)}## which is trivial to integrate.

Any help?

Any help?