Non-elementary integrals

It is a strange fact that many natural looking integrals turn out to defy solution. Students often wonder how they can be done, and are baffled by them. Unlike the case with differentiation, we seem to run into a stone wall very easily while doing integration.


There is a good reason for this bafflment: some of these integrals can be shown to be "non-elementary"! This means they cannot be expressed in terms of the usual functions we know (polynomials, rational functions, trigonometric, logarithmic and exponential functions, and all possible combinations of these). And these claims can actually be proved! Consequently, try as we might, we will not be able to do these integrals in the way we have gotten used to. The only way is to introduce entirely new functions.


Here are some integrals that fit this description: 

• $$ \int e^{x^2} \, dx $$
• $$ \int \frac{\sin x}{x} \, dx $$
• $$ \int \sqrt{\sin x} \, dx $$

On the other hand, there are some definite integrals involving these very same functions which can be done in an elementary way. For example, we have the following very famous identities involving definite integrals: 

• $$ \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi} $$

(this integral arises in the study of the normal distribution in probability theory) and 

• $$ \int_{-\infty}^{\infty} \frac{\sin x}{x} \, dx = \pi. $$

But the anti-derivatives of both $e^{-x^2}$ and $\sin x/x$ are non-elementary! So the proofs of these two identities (and others like them) involve completely different ideas, not involving anti-differentiation.


Here is another such example - a rather spectacular identity (first discovered by one of the Bernoullis, I think): 

• $$ \int_0^1 \frac{1}{x^x} \, dx = \sum_{n=1}^{\infty} \frac{1}{n^n}. $$

The last one may be done using integration by parts, after first writing $x^x$ as $$e^{x \ln x}$$.


Here are some links for those wishing to read more on this topic:

• Non-elementary integral (Wolfram)
• Non-elementary integral (Wikipedia)
• Non-elementary integrals