Integrating x and xdx: What's the Difference?

In summary, when integrating a function, it is important to specify what variable you are integrating with respect to. This is typically done by writing ∫ f(x) dx, where x is the variable. In the case of y = 2x, you can write ∫ 2x dx or ∫ y dx, but not ∫ y dy.
  • #1
Make
20
0
Hello. I'm sort of confused about when you can integrate... Say for example if you have dx you're sort of integrating a "hidden" constant, right? So you increase the power by one and either use the upper and lower limit or just add c: x+c. And similarly if you have xdx, again you increase the power by one and divide:0.5x^2, right? But when are you allowed to integrate? Namely, must you always first obtain dx or dvariable somehow? How do I obtain this in the following case for example: Y=2x? If it is not necessary, then what difference is there in integrating x or xdx? THANK YOU SO MUCH FOR YOUR HELP!
 
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  • #2
Hi Make! :smile:

I'm not sure what you're asking, but does this help? …

When you integrate a function f(x), you have to say what you're integrating it with respect to.

Usually, obviously, it's with respect to x, in which case we write either

"integral of f(x) with respect to x"

or

"∫ f(x) dx".

You can't write an ∫ without a d(something).​

As to your y = 2x example, you can write ∫ 2x dx, or ∫ y dx … they're the same (but they're not the same as ∫ y dy, because that's integrating with respect to something different).
 
  • #3
dead on answer for makes level of understanding :)
 

FAQ: Integrating x and xdx: What's the Difference?

What is the purpose of integrating x and xdx?

Integrating x and xdx is used to find the area under a curve, also known as the definite integral. This is important in many fields of science, such as physics and engineering, as it allows for the calculation of quantities such as displacement, velocity, and acceleration.

What is the difference between integrating x and xdx?

The difference lies in the notation used. When integrating x, we are finding the antiderivative or indefinite integral of x, which is represented as ∫x dx. When integrating xdx, we are finding the definite integral of x, which is represented as ∫xdx.

How do I integrate x and xdx?

To integrate x, you simply add 1 to the exponent and divide by the new exponent. For xdx, you add 1 to the exponent and divide by the new exponent, then multiply by the coefficient of x. For example, the integral of x is 1/2x^2 and the integral of xdx is 1/3x^3.

What are the applications of integrating x and xdx?

Integrating x and xdx has many applications in science and engineering. It is used to determine the work done by a force, the change in energy of a system, and the area under a velocity-time curve to find displacement. It is also used in fields such as economics and biology to model and analyze various phenomena.

What are some common techniques for integrating x and xdx?

Some common techniques for integrating x and xdx include the power rule, substitution, and integration by parts. These techniques allow for the integration of more complex functions by breaking them down into simpler parts. It is important to practice and understand these techniques in order to effectively integrate x and xdx in various applications.

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