integrate: (x^(-1))(e^x)

[Davidmf]

How can I integrate this function?

(x^(-1))(e^x)

A problem like this appeared in my calculus text in the written exercises for the section on integration by parts. I understand in the 'by parts' method, including tabular integration, but the answer is eluding me. I've read in a couple places that this integral isn't solvable without more advanced techniques, but then I wouldn't understand why it is appearing in the 'integration by parts' section in my textbook. I can find the correct answer in the back of the book, and I can differentiate it correctly to get the original problem, but I still don't see how to use integration by parts to get this answer. Can anyone help me with this?

 

[Deleted member 4993]

How can I integrate this function?

(x^(-1))(e^x)

A problem like this appeared in my calculus text in the written exercises for the section on integration by parts. I understand in the 'by parts' method, including tabular integration, but the answer is eluding me. I've read in a couple places that this integral isn't solvable without more advanced techniques, but then I wouldn't understand why it is appearing in the 'integration by parts' section in my textbook. I can find the correct answer in the back of the book, and I can differentiate it correctly to get the original problem, but I still don't see how to use integration by parts to get this answer. Can anyone help me with this?

You are correct - this function can only be integrated as a special function.

I suspect the problem was meant to be \(\displaystyle \int [x * e^x] dx\)

In that case you can use"integration by parts".

 

[daon2]

How can I integrate this function?

(x^(-1))(e^x)

A problem like this appeared in my calculus text in the written exercises for the section on integration by parts. I understand in the 'by parts' method, including tabular integration, but the answer is eluding me. I've read in a couple places that this integral isn't solvable without more advanced techniques, but then I wouldn't understand why it is appearing in the 'integration by parts' section in my textbook. I can find the correct answer in the back of the book, and I can differentiate it correctly to get the original problem, but I still don't see how to use integration by parts to get this answer. Can anyone help me with this?

That function should not have appeared in a calculus textbook. Integration by parts will only make the integral more complicated. If you have seen series, you can integrate the Maclauren series

\(\displaystyle \displaystyle \int \dfrac{1}{x} e^x\, dx = \int \sum_{n=0}^{\infty}\dfrac{x^{n-1}}{n!}\,\, dx = \ln x+\sum_{n=1}^{\infty}\dfrac{x^{n}}{n\cdot n!}+C \)

which has an interesting convergence properties.