# What is the proper verb for "doing" an integral?

It's time to write exams, and when writing in committee we often discover differences in usage between various instructors. Here's an example I noticed today.

What is the proper verb to use in a question of the form: "____ the following integrals"? I've seen evaluate, compute, solve, and others.

• In Russian, there is a specific verb for integrals, and it is, amusingly, "take". Uncomputable integrals are, therefore, named "untakeable" :-) May 15, 2014 at 4:32
• Personally I either find, calculate or integrate them (I might 'find this integral' or 'integrate that term' for example). I also compute or evaluate integrals, particularly definite integrals. I never solve them, unless they're written as an equation involving some unknown (in which case, it's actually the equation that's being solved for the unknown). May 15, 2014 at 4:37
• I proctored some exams this week. A student raised her hand and asked, "what does evaluate mean?" I said "solve" and thought about this question at MESE. May 31, 2014 at 10:34
• I think I would have said, "do" which is how I got here in the first place. :-) Jun 4, 2014 at 19:34
• @KCd I meant "взять интеграл", "неберущиеся интегралы". The verb "интегрировать" means "to integrate", applying to another object (a function). Sep 5, 2016 at 1:43

I'm no native English speaker, but you can tackle that question from the mathematical point of view as well.

The best verb depends on how you view the nature of definite and indefinite integrals.

## Operators/Functionals

Indefinite integrals are operators mapping functions to a set of functions or function (as representative of the equivalence class).

Definite integrals are operators mapping functions to numbers for given intervalls or operators mapping numbers (as one boundary) to numbers for given function and other boundary. In that case, it's numbervalued function of numbers.

Consequently, you should use evaluate.

## Items of a special calculus named "calculus"

The theory "calculus" is a special calculus. Consequently, you should use calculate, even for indefinite integrals.

## Problems

If you see integrals as problems, which non-computable integrals or integrals difficult to compute are indeed, then you should use find.

## Differential equations

Regard the differential equation $\frac{\mathrm{d}F(x)}{\mathrm{d}x}=f(x)$ to be solved for $F(x)$. Its solution is $F(x)=\int f(x)\,\mathrm{d}x$ (or $\{F(x)\}=\int f(x)\,\mathrm{d}x$).

### As solution

If you regard the integral as that solution, you should use the verb associated to solutions of algebraic equations: find.

### As equation itself

If you regard the integral as the equation itself, you should use the verb associated to (algebraic) equations: solve.

• This answer uses English better and is both better explained and justified than mine. +1. Your reasoning I'd use solve $dy/dx=x\sin x$, but still can't bring myself to solve $\int x \sin x dx$. Your introduction of calculate is very good - this is a strong word to use. I often find non-native speakers have thought about language more clearly than I have! May 14, 2014 at 15:18
• I would quibble that the indefinite integral maps functions to functions, not onto functions. But I like the mathematical approach! Can you explain the Problems perspective? I can't parse your sentence there. May 14, 2014 at 15:52
• @MatthewLeingang 1. I have language difficulties with the correct prepositions of to map. If it should be to instead of onto, I'll change it. 2. Take the example $\int\frac{\sin x}{x}\mathrm{d}x$ for someone who doesn't know the sine integral function yet. It's not a simple calculation/operation to determine the integral, as all usual approaches fail. So, this is a real problem, whose solution needs to be found. The less you know about integrals, the more integrands are problematic. So, one could think, that integrals are problematic in general, with many of them already solved. May 14, 2014 at 17:01
• @MatthewLeingang I think what AndrewC may be saying (which I agree with) is that one can only solve an equation, and $\int f(x)\,\mathrm{d}x$ is not an equation. So one shouldn't say "solve $\int f(x)\,\mathrm{d}x$." However you could rephrase it as "solve $\int f(x)\,\mathrm{d}x=F(x)$ for $F(x)$" which represents the same mathematical problem. (oh wait, I just realized this wasn't what you were asking in your comment, but I'll leave this comment here for posterity) May 14, 2014 at 21:15
• @Toscho: Some people use onto as an adjective synonymous with surjective. ("A bijection is a map which is both one-to-one and onto.") For that reason, I only use the preposition onto for surjective maps. So yes, I think it's clearer to say to. May 22, 2014 at 14:54

I would avoid the verb solve as I reserve this for things like equations, inequalities and problems. An integral is equal to a number or a function, so verbs like find, evaluate etc are more appropriate.

I'd use compute only for numerical integration methods.

evaluate and find are the two verbs that are used in textbooks and exams that I've come across. I think evaluate is better for a definite integral because the result is a number, and find is OK for both, eg

Find $\int \sin x e^{\cos x}dx$
or
Find the value of $\int _1^\infty \frac 1 {x^2} dx$
or
Evaluate $\int _2^5 x(2x+1)\sin x dx$

• I prefer evaluate as find makes me thing the answer is under my couch, misplaced somewhere. May 14, 2014 at 14:24
• @JoeTaxpayer That made me laugh! Evaluate is correct and good formal language. May 14, 2014 at 14:27
• I personally would use compute for determining any sort of integral in a calc class outside of a theoretical context. That's what you are doing anyways tough I could see the desire for a more specific term for when doing numerics. However, I'd probably use the term "numerically compute". May 15, 2014 at 1:50
• @JoeTaxpayer That's what it sometimes feels like to me - the answer is somewhere nearby but I haven't quite located it yet, so I find the usage of find for integrals quite apt. May 15, 2014 at 4:41
• @JoeTaxpayer I am reminded of a certain flippant answer that made it's way onto a t-shirt. May 15, 2014 at 13:36

I usually phrase it as "Determine $\int x^2\ dx$" or "Determine $\int_1^3 x^2\ dx$". This way

1) it doesn't tip them off to what type of answer they should arrive at, and

2) it allows them to read the symbol $\int$ as either "integral" or (better in my opinion) "antiderivative".

For completeness, in some problems I write "Set-up, but do not evaluate, the integral that computes some geometrical/physical thing." But when I actually want them to perform the integration (which is another phrase I use colloquially but never as instructions) and arrive at an answer, I write "Use integration to find the geometrical/physical thing."

• Could you explain how determine is better than find or evaluate in 2? I mean, do other choices preclude them from reading the integral sign as "antiderivative"? May 14, 2014 at 15:48
• Personally I think of "evaluate" as resulting in a numerical value. And for me "find" is more appropriate for the setting up of formulas/equations. My point in 2) was that I avoid using the word "integral" in the way that the OP used it. May 14, 2014 at 15:58
• @MatthewLeingang determine is completely general. It doesn't say anything about the nature of the integral and doesn't fix any way of solution. In that case, you need to accept, if students guess the integral somehow and proof it by its derivative. May 14, 2014 at 17:03

The proper verb is whatever verb your textbook uses.

This helped me one time when I wrote "Evaluate this integral", and one student approximated the integral using rectangles. I was able to show the student that every problem I assigned from the textbook that required a symbolic answer said "Evaluate this integral" whereas every problem that required a numerical approximation said "Approximate this integral". The student did not argue.

This is a good policy for any exam problem, not just integrals. I always use the same phrasing on exams that my textbook uses.

• This is a good point. At my place of work our calculus courses are coordinated with several instructors but the same final. I push for consistency with textbook notation, not because it's best, but because it's common. May 21, 2014 at 11:05

From a general test writing standpoint: If you and the students both know that the problems are to be solved by integrating a formula, why not "Integrate the following..." If they have to figure out whether to use integration or some other calculus then that won't work.

In practical applications, such as HPLC analysis, the usual instruction is calculate the amount of substance X by integrating the peak area in the chromatogram in comparison to a standard chromatogram