# Is this integral unsolvable?

So I took an integration test in AP Calculus yesterday and everything went smoothly except for one question. $$\int \frac{e^x}{x^2}dx$$ I tried chain rule, $$u$$ substitution, and all methods we have since been taught. Nothing worked. Now, I really wanted to know what the answer was, so I just now plugged it into a few antiderivative/integral calculator and all but two said something along the lines of "cannot solve." One said $$\frac{exp(x)}{x^2} + C$$. The other said $$Ei(x) - \frac{e^x}{x} + C$$. A physics forum about the same integration seemed to say that it is unsolvable. I am thoroughly confused. Can anybody help clarify?

Edit: The teacher made a little error. The real question was supposed to be, which is of course much easier: $$\int \frac{e^\frac{1}{x}}{x^2}$$

• Welcome to Math Stack Exchange. Don't forget $dx$ ! – J. W. Tanner Mar 12 at 0:33
• It cannot be expressed with the elementary functions. – Bernard Mar 12 at 0:36
• @RileyFitzpatrick Are you sure this was the original question? Maybe you were supposed to approximate the integral. – Toby Mak Mar 12 at 0:44
• I am positive this is the original question. I am in high school, so the goal for this section was really just to find the antiderivative. Most of them were very simple. I only remember so clearly because it confused me so much. Do you think my teacher just made an error? – Riley Fitzpatrick Mar 12 at 0:55
• Sounds like your teacher just made an error. – Qiaochu Yuan Mar 12 at 1:53

## 1 Answer

$$\operatorname{Ei}(x)-e^x/x + C$$ is right -- at least on each side of $$0$$ separately -- but you may not really consider it a "solution" of the kind you were hoping for, because $$\operatorname{Ei}$$ stands for the exponential integral, which is defined as $$\operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} dt$$ (and you can then verify that $$\operatorname{Ei}(x)-e^x/x$$ has the derivative you want). So this solution is just swapping one integral without an elementary solution for the other.

It is still a bit of progress, though, since it is easier to find facts about the exponential integral than about your version.

In practice, one would evaluate the integral numerically.

• Great, this helps. So based on the fact that I am in high school and all of the adjacent questions were simple antiderivatives ($+C$), do you think my teacher made a mistake? – Riley Fitzpatrick Mar 12 at 0:56
• @RileyFitzpatrick Perhaps your teacher wanted to teach you a valuable lesson about how integrating is hard :) – Jair Taylor Mar 12 at 1:01