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