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: $$\int \frac{e^\frac{1}{x}}{x^2}$$

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    $\begingroup$ Welcome to Math Stack Exchange. Don't forget $dx$ ! $\endgroup$ Mar 12 '19 at 0:33
  • $\begingroup$ It cannot be expressed with the elementary functions. $\endgroup$
    – Bernard
    Mar 12 '19 at 0:36
  • $\begingroup$ @RileyFitzpatrick Are you sure this was the original question? Maybe you were supposed to approximate the integral. $\endgroup$
    – Toby Mak
    Mar 12 '19 at 0:44
  • $\begingroup$ 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? $\endgroup$ Mar 12 '19 at 0:55
  • $\begingroup$ Sounds like your teacher just made an error. $\endgroup$ Mar 12 '19 at 1:53

$\operatorname{Ei}(x)-e^x/x + C$ 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.

  • $\begingroup$ 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? $\endgroup$ Mar 12 '19 at 0:56
  • $\begingroup$ @RileyFitzpatrick Perhaps your teacher wanted to teach you a valuable lesson about how integrating is hard :) $\endgroup$ Mar 12 '19 at 1:01

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