Numerical integration is a standard topic in first-semester calculus. From time to time, I have received questions from students on various aspects of this topic, including:

- Why is numerical integration necessary in the first place?
- Where do these formulas come from (especially Simpson’s Rule)?
- How can I do all of these formulas quickly?
- Is there a reason why the Midpoint Rule is better than the Trapezoid Rule?
- Is there a reason why both the Midpoint Rule and the Trapezoid Rule converge quadratically?
- Is there a reason why Simpson’s Rule converges like the fourth power of the number of subintervals?

In this series, I hope to answer these questions. While these are standard questions in a introductory college course in numerical analysis, and full and rigorous proofs can be found on Wikipedia and Mathworld, I will approach these questions from the point of view of a bright student who is currently enrolled in calculus and hasn’t yet taken real analysis or numerical analysis.

In this post, I’d like to take a closer look at the indefinite integral , which is closely related to the area under the bell curve . This integral cannot be computed using elementary functions. However, using integration by parts, there are some related integrals that can be computed:

Based on these examples, it stands to reason that, if can be written in terms of elementary functions, it should have the form

,

where is some polynomial to be determined. We will now show that this is impossible.

Suppose , a polynomial of degree to be determined. Then we have

or

or

.

In other words, all terms on the left-hand side except the constant term must cancel. However, this is impossible: is a polynomial of degree while is a polynomial of degree . Therefore, the left hand side must have degree and therefore cannot be a constant.

A similar argument shows that cannot have the form , where the exponents may or may not be integers.

This may be enough to convince a calculus student that there is no elementary antiderivative of . Indeed, although the proof goes well beyond first-year calculus, there is a theorem that says that if can be expressed in terms of elementary functions, then the antiderivative must have the form . So the guess above actually can be rigorously justified. References:

- Elena Anne Marchisotto and Gholam-Ali Zakeri, “An Invitation to Integration in Finite Terms,”
*The College Mathematics Journal*, Sep., 1994, Vol. 25, No. 4 (Sep., 1994), pp. 295-308 - J. F. Ritt,
**Integration in Finite Terms: Liouville’s Theory of Elementary Methods**, Columbia University Press, New York, 1948

This is great !