Is it possible for integrals to have functions of the integrand in the bounds of the integral such as $\int_0^x xdx$?

Question

Is it possible for integrals to have functions of the integrand in the bounds of the integral such as $\int_0^x x\,dx$ or more generally $\int_0^{f(x)} x\,dx$ ?

I have seen this several times in the past few weeks for the first time and I am not sure whether they were typos. If this is possible, would the solution be :

$$\left. \frac {x^2} 2 \right|^x_0 \text{ ?}$$

People sometimes write things like $\displaystyle \int_0^x \cos x,dx$ when they are in effect treating the $x$ in $\displaystyle \int_0^x$ and the $x$ in $\cos x,dx$ as two different things. It is at best infelicitous notation and I avoid it. $\qquad$ — Michael Hardy, Aug 03 '17 at 02:24
https://math.stackexchange.com/questions/3533541/integral-notation-confusion-x-vs-t?noredirect=1 — Mike Earnest, Sep 13 '23 at 17:48

score 2 · Accepted Answer · answered Aug 03 '17 at 01:22

2

No. The variable of integration is a loop variable and it goes out of scope before the endpoints get evaluated.

answered Aug 03 '17 at 01:22

ncmathsadist

49,383

1

I live in two worlds. Scope of variables should be taught to mathematicians. – ncmathsadist Aug 03 '17 at 01:24
Are you saying no to the first question of the possibility of evaluation or to the proposed solution? – IntegrateThis Aug 03 '17 at 01:25
2

It simply makes no sense. – ncmathsadist Aug 03 '17 at 01:26
5

It's sometimes used as bad notation - the variable of integration should be a dummy variable that doesn't appear outside the integral term. – ConMan Aug 03 '17 at 01:54

Is it possible for integrals to have functions of the integrand in the bounds of the integral such as $\int_0^x xdx$?

1 Answers1