Is there a standard way to write a second definite integral?

This question arose in the context of writing the equations of kinematics for a point mass with constant acceleration. The integration parameter $t$ represents time.

This is so basic that I didn't even realize I didn't know how to do it. I define $\Delta t$ as the definite integral of $1$ from $t_0$ to $t$ as shown. I then integrate $\Delta t$ from $t_0$ to $t$ producing the desired result.

$$\begin{aligned} \int_{t_{0}}^{t}dt=\Delta t&=t-t_{0}\\ \int_{t_{0}}^{t}\Delta tdt&=\int_{t_{0}}^{t}\left(t-t_{0}\right)dt\\ &=\frac{1}{2}t^{2}-tt_{0}-\left(\frac{1}{2}t_{0}^{2}-t_{0}^{2}\right)\\ &=\frac{1}{2}\left(t-t_{0}\right)^{2}\\ &=\frac{1}{2}\Delta t^{2} \end{aligned}$$

I might write that as $$\int_{t_{0}}^{t}\left(\int_{t_{0}}^{t}dt\right)dt=\int_{t_{0}}^{t}\int_{t_{0}}^{t}dt^{2}=\frac{1}{2}\Delta t^{2}.$$

In that case, the upper bound of integration of the inner integral is the value of the integration variable of the outer integral, so $t$ has two meanings.

To show why I find this problematic, I will make the upper bound of integration more explicit. Suppose I treat the upper bound of integration as a specific fixed value $t_{1}$. Then the inner integral behaves as a constant with respect to the second integral, and I get a different result.

$$\begin{aligned} \int_{t_{0}}^{t_{1}}dt=\Delta t&=t_{1}-t_{0}\\ \int_{t_{0}}^{t_{1}}\int_{t_{0}}^{t_{1}}dt^{2}&=\int_{t_{0}}^{t_{1}}\Delta tdt\\ &=\int_{t_{0}}^{t_{1}}\left(t_{1}-t_{0}\right)dt\\ &=\Delta t\int_{t_{0}}^{t_{1}}dt\\ &=\Delta t^{2} \end{aligned}$$

One remedy I might propose is to write the bound of the outer integral as $t_{1}$ and leave the bound of the inner integral as the dependent variable $t$.

$$\begin{aligned} \int_{t_{0}}^{t_{1}}dt=\Delta t&=t_{1}-t_{0}\\ \int_{t_{0}}^{t_{1}}\int_{t_{0}}^{t}dt^{2} &=\int_{t_{0}}^{t_{1}}\left(t-t_{0}\right)dt\\ &=\frac{1}{2}\Delta t^{2} \end{aligned}$$

It appears to communicate my intent, but I don't recall having encountered that use of notation in the case of a second integral (as opposed to the more general case of a double integral). Where I have encountered such integrals in textbooks, the authors typically speak of antiderivatives and constants of integration, etc., or sidestep the entire problem by writing two separate subsequent integrals.

Is there a standard way of writing what I am here calling second definite integral?

Solution 1:

Just make the two variables of integration different: $$ \int_{t_0}^{t_1} \int_{t_0}^u dt\,du $$

This is a consequence of the general idea that to minimize confusion the variable of integration should not be chosen as a letter that is already bound to a value outside of the integral. You're breaking this principle already at the beginning when you write $$ \int_{t_0}^t dt$$

(Yes, this is in contrast to the Leibniz notation for derivatives, where the same variable letter is used inside and outside the $\frac{d}{dx}$ operator. But the notation we have is the notation we have, and trying to straighten it out is usually not worth the risk of confusion).