Setting the Wrong Boundaries: Why Definite Integral Substitution Goes Wrong

This one is built for substitution: put $u = x^2+1$, so $du = 2x\,dx$, and the $2x\,dx$ already inside becomes $du$ exactly. Here is the careful-looking version that goes wrong:

Every symbol inside the integral was converted; the two numbers outside it were copied from the line above, and now name the wrong variable.

A definite integral is a sum accumulated along an axis: $\int_{a}^{b} f(x)\,dx$ says start at $x = a$ and finish at $x = b$. Change the variable and you change the axis you walk along, so both endpoints need renaming. Under $u = x^2+1$, the point $x = 1$ is the point $u = 2$: same place, new label. The far endpoint $x = 2$ is $u = 5$:

One line covers every substitution, not just this one:

The new endpoints are the old ones sent through $g$: here $g(1) = 2$ and $g(2) = 5$, while under $u = \sin x$, $x = \pi/2$ becomes $u = 1$.

A rough estimate catches $1.219$: over this one-unit interval the integrand never dips below $2\sqrt{2}$.

Keeping the old limits is legal only if you return to $x$ first: $\frac{2}{3}(x^2+1)^{3/2}$ from $1$ to $2$ gives the same $5.568$. Never evaluate a $u$-expression at $x$-numbers.

This slip gets filed under carelessness, but research on how students hold the integral in mind points somewhere more specific. Asked in open questions what a definite integral means, 150 calculus students at two US colleges mostly reached for the area under a curve or the antiderivative; answers describing it as adding up many small pieces came up much less often^[1]. Interviews with experienced calculus students turned up several distinct ways of reading the integral sign, and the adding-up-pieces reading looked most useful on problems with several variables and on physics problems^[2]. Nor is this new: in interviews with 100 students aged 16 to 22, questions demanding that sum-across-an-interval reading were hard even for the stronger students^[3].

None of those studies looked at substitution or at the limits of integration^[1][2][3]; the link to this slip is my reading, not theirs. If the integral lives in your head as a picture, or as a template — find the antiderivative, plug in the top, subtract the bottom — then the two numbers on the sign are slots to fill, not the two ends of a walk along the $x$-axis. Slots do not belong to a variable, so nothing warns you when the variable underneath them changes. A second reason, equally untested: the habit is trained. On indefinite integrals you return to $x$ before anything else happens, so the limits never come up while you are still in $u$, and nothing there punishes the reflex.

Sometimes copying the limits gives the right answer, which is worse than getting it wrong. In $\int_{x=0}^{x=1} 2x^{3}\,dx$, write the integrand as $x^2 \cdot 2x$ and let $u = x^2$; the converted limits are $u = 0$ and $u = 1$ — the same two numbers as before, so careful and careless both write $\int_{0}^{1} u\,du = \frac{1}{2}$, and both are right. They agree only because this substitution fixes both endpoints: $0^2 = 0$ and $1^2 = 1$. Passing a problem like this is how the habit survives.

Let $u = x + 2$, so $du = dx$, $x = -2$ gives $u = 0$, and $x = 1$ gives $u = 3$. Then $\int_{u=0}^{u=3} u\,du = \left.\frac{u^2}{2}\right|_{0}^{3} = 4.5$, while the old limits would give $\int_{u=-2}^{u=1} u\,du = \frac{1}{2}(1 - 4) = -1.5$.

That is not merely off, it is impossible: $x + 2$ is never negative here, so the integral cannot be. The region is a triangle with base $3$ and height $3$: area $4.5$.

The wrong solution above is printed in full on purpose. In two experiments run in real algebra classrooms, the first on a unit in solving two-step linear equations, students who explained worked examples alongside their usual practice understood the material better than those who only practised^[4]. The second experiment suggested wrong examples, alone or paired with correct ones, may be the most useful kind for that.

A smaller study points the same way. Nine- to 11-year-olds learning about decimals who compared a worked-out wrong answer with a correct one grasped the method and the ideas behind it better than classmates who compared two correct answers, and held fewer wrong ideas — a brief session with children, far from calculus^[5].

Saying why out loud has its own evidence. A review that gathered the studies on prompting students to explain their own steps found a modest boost on tests taken straight afterwards, to both doing the mathematics and understanding it, and a bigger one where students were helped to produce good explanations; evidence that it works in ordinary classrooms, or that the gain lasts, was thinner^[6].

So make one spoken sentence your check the moment you finish substituting: these two numbers — which variable do they belong to? If the answer is $x$ and you are integrating in $u$, convert them. In two studies with 10- to 12-year-olds learning about decimals, understanding and skill grew together, and more help in seeing what a problem was asking improved skill^[7]. Children on decimals are not calculus students, so take it only as a hint: the cure for a dropped step is probably not twenty more repetitions.

Evaluate $\int_{x=0}^{x=3} 6x\left(x^2+1\right)^{2} dx$. Substitute, convert both limits, evaluate; cover the next paragraph until you have a number.

Let $u = x^2+1$, so $6x\,dx = 3\,du$; $x = 0$ gives $u = 1$ and $x = 3$ gives $u = 10$. Then $\int_{u=1}^{u=10} 3u^{2}\,du = \left.u^{3}\right|_{1}^{10} = 999$, where the old limits give $27$. Expanding instead confirms it: $6x(x^2+1)^2 = 6x^5 + 12x^3 + 6x$, with antiderivative $x^6 + 3x^4 + 3x^2$, which is $999$ at $x = 3$ and $0$ at $x = 0$.

Not one of the studies behind this post looked at substitution, at changing variables, or at the limits of integration^{[1][2][3][7][6][5][4]}. Separate studies decades apart do point the same way: calculus students mostly did not explain the integral as a sum across an interval^[1], and questions demanding that reading were hard even for the stronger students^[3]. The step from there to and that is why the limits get copied is mine, and untested. No study here counts how often this error happens.

The advice half is thinner than it looks. The closest study to a reader here is the pair of algebra classroom experiments^[4]; the other two behind the wrong-example advice are 9- to 12-year-olds working on decimals^[7][5]. One point deserves saying plainly: prompting learners to explain why a common wrong method is wrong is a recommendation the review's authors make from the wider literature, not a result that review measured; the same review found the evidence for classroom or lasting gains weak^[6]. None of the mathematics above depends on any of this; it is proved on the page.