Why is integral notation sometimes written in a strange way?

[amcavoy]

This is a strange question, but why (in some physics books) are integrals given as:

$$\int dx\,f\left(x\right)$$

rather than

$$\int f\left(x\right)\,dx$$

??

 

[Dr Transport]

No real reason, they are equivalent statements...

 

[sir-pinski]

The notation \int dx\,f\left(x\right) is actually not strange at all and is commonly used in mathematics and physics. It is called the Leibniz notation, named after the mathematician Gottfried Leibniz who introduced it in the 17th century.

The reason for this notation is to emphasize the relationship between the integral sign and the function being integrated. In this notation, the "dx" is treated as a variable, just like "x" in the function f(x). This highlights the fact that the integral is essentially a sum of infinitely small elements, with "dx" representing the infinitesimal width of each element.

Furthermore, the Leibniz notation makes it easier to understand and manipulate integrals using the rules of differentiation. For example, the notation \int dx\,f\left(x\right) can be seen as a reverse of the derivative notation \frac{df}{dx}, where the "dx" is moved to the other side of the fraction. This makes it easier to apply techniques like integration by substitution.

On the other hand, the notation \int f\left(x\right)\,dx, known as the Riemann notation, is more commonly used in introductory calculus courses. It is simpler and easier to understand for beginners, as the "dx" is treated as a separate term outside the integral sign.

In short, the Leibniz notation \int dx\,f\left(x\right) may seem strange to some, but it is a useful and widely accepted notation in mathematics and physics. Its purpose is to emphasize the relationship between the integral and the function being integrated, and to make it easier to manipulate and apply techniques of differentiation.