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How do you integrate #(e^x/x)dx #?

Answer 1

Emily Ammerman

This is sometimes called the exponential integral:

#inte^x/xdx="Ei"(x)+C#

But the method I'd use (since I'm not familiar with the integral) is the Maclaurin series for #e^x#:

#e^x=1+x+x^2/(2!)+x^3/(3!)+...=sum_(n=0)^oox^n/(n!)#

Then:

#e^x/x=1/x+1+x/(2!)+x^2/(3!)+...=1/x+sum_(n=0)^oox^n/((n+1)!)#

So the antiderivative will be:

#inte^x/xdx=int(1/x+1+x/(2!)+x^2/(3!)+...)dx=ln(absx)+x+x^2/(2*2!)+x^3/(3*3!)+...+C#

#inte^x/xdx=ln(absx)+sum_(n=1)^oox^n/(n*n!)+C#

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Answer 2

Evelyn Agard

To integrate ( \frac{e^x}{x} , dx ), you can use the Exponential Integral function, denoted as ( \text{Ei}(x) ). The integral does not have a simple closed-form expression using elementary functions. Therefore, it is expressed using a special function, the Exponential Integral. The integral can be represented as ( \text{Ei}(x) + C ), where ( C ) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.