# integration$\int \frac{e^x}{x}dx$

I want to integrate $\int \frac{e^x}{x}dx$.

Attempt 1: Let $u = e^x$, and $dv = \frac{1}{x}dx$ i.e. $v = \log x$. Then by integration by parts, we have \begin{align} \int \frac{e^x}{x}dx &= \int u\, dv \\ &= uv - \int v\, du \\ &= e^x\log x - \underbrace{\int e^x\log x\,dx}_{=?} \end{align}

Attempt 2: Let $u = \frac{1}{x}$ and $dv = e^xdx$. Then similarly we have \begin{align} \int \frac{e^x}{x}dx &= uv - \int v\, du \\ &=\frac{e^x}{x} - \int e^x\Big(-\frac{1}{x^2}\Big)dx \\ &=\frac{e^x}{x} + \underbrace{\int\frac{e^x}{x^2}dx}_{=?} \end{align}

I tried choosing different $u$s by using integration by parts, but none worked

Does anyone have a better idea?

• This is not an elementary integral. Check the exponential integral. – Mhenni Benghorbal Jan 6 '15 at 8:05
• I don't think this is a very simple integral to evaluate, but what you have done is correct. – Bman72 Jan 6 '15 at 8:05
• It would be nice if you would try to learn mathjax, see help page, and not to expect us to read your not too excellent handwriting. – Karl Jan 6 '15 at 8:34
• sorry about my awkward handwriting ,as my native language is not English , it is not easy for me to surf through the whole website , I will try using what you called mathjax next time ,thank you – user143997 Jan 6 '15 at 8:55
• @user143997 here is a link that you can use to learn the fundamental of mathjax (/latex) – Bman72 Jan 6 '15 at 10:11

This is a textbook example of an integral that can't be expressed with traditional set of analytical functions. With a specific choice of the integrating constant, you get the exponential integral $\operatorname{Ei}(x)$.