In the indefinite integral, what’s the integration of (e^x)/x ?

Question

Arun · Answer

(e^x)/x has no elementary derivative -- you can try integration by parts but you won't get anywhere.

so mathematicians have made up a special function called the exponential integral, Ei(x), which is defined as:

Ei(x) = integral, from -infinity to x, of (e^t / t) dt

You cannot solve this explicitly (analytically).

However, you can do it in terms of power series!

Recall that
e^x = ∑ xⁿ/n!,so
e^x / x = 1/x + ∑ xⁿ / (n+1)!.

Therefore,
∫e^x / x dx = ln|x| + ∑ xⁿ / (n·n!) + C.

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In the indefinite integral, what’s the integration of (e^x)/x ?

In the indefinite integral, what’s the integration of (e^x)/x ?

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