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

1 Answer
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 \ e^x/x \ 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!) #