Integral of e^x/(x+1) | Solving Methods & Hints for Homework

[Jalo]

Homework Statement

Integral of e^x / (x+1)

Homework Equations

The Attempt at a Solution

I've tried to substitute e^x=c, dx=1/c but it leads me nowhere.
Integration by parts has failed me too... Some hints on how to solve it would be awesome
thanks

 

[D H]

You will not be able to solve this by elementary techniques.

You should be able to convert this to a constant times $\int\frac{e^u}u du$.

You won't be able to solve that by elementary techniques, either. $\int\frac{e^u}u du$ is the exponential integral. It comes up a lot, so it is given a name, the special function Ei(x).

 

[Ray Vickson]

Homework Statement

Integral of e^x / (x+1)

Homework Equations

The Attempt at a Solution

I've tried to substitute e^x=c, dx=1/c but it leads me nowhere.
Integration by parts has failed me too... Some hints on how to solve it would be awesome
thanks

To expand on the response of DS: it is a *Theorem* that the function f(x) = exp(x)/x has *no* elementary anti-derivative. It is not the case that nobody has been smart enough to figure out what the anti-derivative is; instead, it is provably _impossible_ to write down any finite formula--involving only elementary functions--that will give the indefinite integral. So, if you converted every electron in the universe into a large sheet of paper and wrote hundreds of symbols on each page, you still would not be able to write down the antiderivative! This fascinating topic got started by work of Liouville in the late 19th century, and is discussed widely in books and papers on symbolic integration (as in Mathematica or Maple). Still, the integral appears in many applications, so we just invent it as a new function and figure out ways to evaluate it accurately and efficiently.

RGV