# Integration by parts wrong (?)

1. Aug 3, 2011

Here I used integration by parts to try to solve an integral (I got it wrong, it seems), I know this has no "simple" solution, but, can anyone explain me exactly what did I do wrong? Here is what I did:

$\int\frac{e^x}{x}dx=\frac{e^x}{x}-(-1)\int\frac{e^x}{x^2}dx=\frac{e^x}{x}+1(\frac{e^x}{x^2}-(-2)\int\frac{e^x}{x^3}dx)=\frac{e^x}{x}+1!(\frac{e^x}{x^2}+2!\frac{e^x}{x^3}+3!\int\frac{e^x}{x^4}dx)=...=0!\frac{e^x}{x}+1!\frac{e^x}{x^2}+2!\frac{e^x}{x^3}+3!\frac{e^x}{x^4}+...$

I just used integration by parts on the "remaining" integrals so I could produce a series, but the series is just...wrong. I would appreciate if you told me what is the error in my reasoning.
thanks.

2. Aug 4, 2011

### tiny-tim

Last edited by a moderator: Apr 26, 2017
3. Aug 4, 2011

It doesn't, that's the point. Why can I use integration by parts and "say" that $\int\frac{e^x}{x}dx$ equals a sum that does not converge?

4. Aug 4, 2011

And, thanks :)

5. Aug 5, 2011

### tiny-tim

integration by parts works fine so long as the ∑ is finite …

∫ ex/x dx

= ex(∑n=0k-1 n!/xn+1 - n! ∫ ex/xn+1 dx​

unfortunately, although we can usually rely on the final integral converging to zero as k -> ∞, in this case it doesn't!

6. Aug 6, 2011