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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  ?

Grade:12th pass

1 Answers

Arun
25763 Points
3 years ago
(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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