Question

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

Answer 1

`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!)`