# What exactly is dx in an integral

• A.J.710
In summary, dx is the notation for the differential of x in integration by parts. It is important to include it in more complex integrals to avoid running into problems. It is not the same as the derivative of x, which is simply 1.
A.J.710
I'm doing integration by parts and I am a bit confused as to what exactly dx is. Usually when integrating it is just dropped or forgotten about. Now when doing integration by parts there are some problems where you pick x as your u and dx as your du. since du is the derivative of u why isn't the du of x just 1 like we were always taught? Why is it now dx?

A.J.710 said:
I'm doing integration by parts and I am a bit confused as to what exactly dx is.
Mostly it indicates the variable with respect to which you're integrating. It also is suggestive of the Δx in a Riemann sum.
A.J.710 said:
Usually when integrating it is just dropped or forgotten about.
That's not a good thing to do. dx doesn't play much of a role for very simple integrals such as substitutions. However, in integration by parts of trig substitutions, if you omit it, you will run into problems.
A.J.710 said:
Now when doing integration by parts there are some problems where you pick x as your u and dx as your du. since du is the derivative of u
No, du is the differential of u. That's not the same as the derivative.
A.J.710 said:
why isn't the du of x just 1 like we were always taught?
I doubt that's what you were taught. "du of x" makes no sense.
The derivative, with respect to x, of x is 1. In symbols, ##\frac{d}{dx}x = 1##, but the differential of x is dx.
A.J.710 said:
Why is it now dx?

## 1. What does the "dx" represent in an integral?

The "dx" in an integral represents an infinitesimal change in the variable of integration, which is typically denoted as "x". It is used to indicate that the integral is being evaluated with respect to that specific variable.

## 2. Why is "dx" used in integrals?

"dx" is used in integrals to represent the tiny intervals or slices along the x-axis that are being summed up to find the total area under a curve. It helps to show that the integral is being evaluated with respect to the variable of integration.

## 3. Can "dx" be replaced with another letter?

Yes, "dx" can be replaced with any other letter or symbol to represent the variable of integration. This is commonly done when dealing with multi-variable integrals, where different variables are used for each integral.

## 4. Is "dx" a variable in itself?

No, "dx" is not a variable in itself. It is used as a notation to represent the variable of integration in an integral. It is important to understand that the integral is not just a sum of infinitely many "dx" terms, but rather a single quantity representing the total area under the curve.

## 5. How is "dx" related to the derivative?

"dx" and the derivative are closely related, as "dx" represents an infinitesimal change in the variable of integration, while the derivative represents the instantaneous rate of change of a function with respect to that same variable. In fact, the fundamental theorem of calculus states that the derivative and the integral are inverse operations of each other.

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