# What is the antiderivative of (e^x)/x?

Apr 2, 2017

 int \ e^x/x \ dx = lnAx + x + x^2/(2*2!) + x^3/(3*3!) + ... x^n/(n*n!) + ...

or in sigma notation

 int \ e^x/x \ dx = lnAx + sum_(n=1)^oo x^n/(n*n!)

#### Explanation:

Let:

$I = \int \setminus {e}^{x} / x \setminus \mathrm{dx}$

This does not have an elementary solution. Definite integrals involving this integrand are calculated using tables of the Exponential integral

The best you can get is a power series which we derive from the power series of ${e}^{x}$

 e^x = 1 + x + x^2/(2!) + x^3/(3!) + ... x^n/(n!) + ... + ...

So the integral becomes:

 I = int \ 1/x{1 + x + x^2/(2!) + x^3/(3!) + ... x^n/(n!) + ... + ... } \ dx
 \ \ = int \ 1/x + 1 + x/(2!) + x^2/(3!) + ... x^(n-1)/(n!) + ... + ... \ dx
 \ \ = lnx + x + x^2/(2*2!) + x^3/(3*3!) + ... x^n/(n*n!) + ... + C
 \ \ = lnAx + x + x^2/(2*2!) + x^3/(3*3!) + ... x^n/(n*n!) + ...
 \ \ = lnAx + sum_(n=1)^oo x^n/(n*n!)