$\int e^{x} / x$

Editor: This integral often crops up in Ask NRICH. Similar integrals also frequently arise which can be converted via integration by parts and substitution into this form are

\int e^{x} \log x

and

\int \log x / e^{x}

By Arun Iyer (P4587) on Friday, June 08, 2001 - 07:48 pm:

What is the value of

the indefinite integral

\int e^{x} / x

By Brad Rodgers (P1930) on Friday, June 08, 2001 - 11:17 pm:

The answer is essentially, that there is no answer. The integral of this function cannot be expressed by any combination of algebraic functions (though I'm afraid I don't really know how to prove this). Instead, mathematicians invented the function

Ei (x)

, defining it as

$Ei (x) = γ + \ln (x) + \sum_{n = 1}^{\infty} x^{n} / (n! n)$

(with the $γ$ on there for convenience in other calculations; $γ = lim_{n \to \infty}] (\sum_{k = 1}^{n} (1 / k) - \ln (n))$ )

To show that $\int e^{x} / x = Ei (x)$ , we want to show that $d / dx (Ei (x)) = e^{x} / x$ .

So,

$d / dx (Ei (x)) = d / dx (γ) + d / dx (\ln (x)) + d / dx (\sum_{n = 1}^{\infty} x^{n} / (n! \times n)$

$= 1 / x + 1 + x / 2! + x^{2} / 3! + x^{3} / 4!$

(i.e.

$\sum_{n = 1}^{\infty} x^{n} / (n! \times n) = x + x^{2} / (2!) + x^{3} / (3! \times 3) + x^{4} / (4! \times 4) + \dots$

So,

$d / dx (\sum_{n = 1}^{\infty} x^{n} / (n! \times n)) = d / dx (x) + d / dx (x^{2} / 2! \times 2) + d / dx (x^{3} / 3! \times 3) + \dots)$

As, recalling the power series for $e^{x}$ ,

$e^{x} / x = 1 / x + 1 + x / 2! + x^{2} / 3! + \dots$

we now have shown that $d / dx (Ei (x)) = e^{x} / x$ , as required.

If there's anything you don't understand, don't hesitate to write again,

Brad

By Arun Iyer (P4587) on Thursday, June 14, 2001 - 07:47 pm:

Thanks Brad,
I think I get the picture that you are trying to show me.
Arun