# What's the variable of integration?

• SammyS
In summary, the conversation discusses the ambiguity of the variable of integration in a given integral expression. It highlights the importance of clarifying the variable of integration and the potential for different interpretations depending on the context.

#### SammyS

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What's the variable of integration?

I keep seeing posts like this:
Help me do this integral

$$\int xe^{-ax}$$

Shouldn't we all ask, right off the bat, "What's the variable of integration?"

Probably what was meant was:

$$\int xe^{-ax}\, dx \,.$$

But maybe the problem actually was:

$$\int xe^{-ax}\, da \,.$$

Does this bother any of you?

Although it is rather clear what was meant (by the convention of using x for a variable and a for a constant) I agree that the notation is not completely unambiguous.
Depending on the context, it would be important or nit-picky (is that a word?) to point this out to the one who gave you that expression.

I am reminded of an old physics professor who, when using a dummy variable x for integration of the real variable x, would sheepishly look around the room checking for any math police.

And if you are really brave, and this were a homework problem, you might just write down
$$\int x e^{ax} = e^{ax} + c$$
(don't forget the + c ) and hand it in like that :)

As a scientist, the variable of integration is an important concept in calculus and mathematical modeling. The variable of integration refers to the independent variable that is being integrated over in a given function or equation. In the example given, the variable of integration is x, as it is the independent variable being integrated over in the function xe^{-ax}.

It is important to clarify the variable of integration in order to correctly set up and solve a given integral. In the given example, the integral is set up as \int xe^{-ax}\, dx, indicating that x is the variable of integration and dx represents an infinitesimal change in x. If the integral were set up as \int xe^{-ax}\, da, then a would be the variable of integration and da would represent an infinitesimal change in a.

It is crucial to properly identify the variable of integration in order to accurately solve integrals and perform mathematical modeling. This ensures that the correct independent variable is being integrated over and that the resulting solution is meaningful and applicable to the problem at hand. Therefore, it is important to always clarify and double check the variable of integration when working with integrals.

## 1. What is the variable of integration?

The variable of integration is the independent variable that is used in the integration process. It is typically denoted by the letter "x" and represents the quantity that is being summed over in the integral.

## 2. Why is the variable of integration important?

The variable of integration is important because it determines the limits of integration and the direction of integration. It also allows for the evaluation of the integral with respect to a specific quantity.

## 3. How do you determine the variable of integration?

The variable of integration is usually determined by the problem at hand. It is often the same variable that is used in the integrand, but it can also be changed to a different variable for ease of calculation.

## 4. Can the variable of integration be changed during the integration process?

Yes, the variable of integration can be changed during the integration process. This is known as a variable substitution, where a new variable is introduced to simplify the integral. However, the limits of integration must also be adjusted accordingly.

## 5. What happens if the wrong variable of integration is chosen?

If the wrong variable of integration is chosen, the integral may not be solvable or the incorrect answer may be obtained. It is important to carefully choose the variable of integration to ensure the correct evaluation of the integral.