# Integration by Parts: Understanding dv & dx

• Bashyboy
In summary, integration by parts is the integral version of the product rule for derivatives and can be written as udv = d(uv) - vdu. When integrating both sides, we get \int udv = \int d(uv) - \int vdu, with \int d(uv) = uv. The reason for including dx in dv is because dv is a differential and cannot be equal to an ordinary function. This is a defined rule in mathematics.

#### Bashyboy

I understand this integration technique, for the most part. One thing I am curious to know is why, when you do your rudimentary substitution for this particular technique, does dv have to always include dx?

"Integration by parts" is just the integral version of the "product rule" for derivatives:
d(uv)= udv+ vdu. We can write that as udv= d(uv)- vdu and integrate both sides:
$\int udv= \int d(uv)- \int vdu$. Of course, $\int d(uv)= uv$.

As for "does dv have to always include dx?", yes, of course. "dv" is a differential and you cannot have an "ordinary" function equal to a differential. A differential can only be equal to another differential.

HallsofIvy said:
As for "does dv have to always include dx?", yes, of course. "dv" is a differential and you cannot have an "ordinary" function equal to a differential. A differential can only be equal to another differential.

Is there a reason for this? Or have mathematicians defined this to be true?

## What is Integration by Parts?

Integration by Parts is a method used to evaluate integrals that involve products of functions. It involves rewriting the integral in a different form in order to make it easier to solve.

## How do you choose which function to use for "dv" and "dx" in Integration by Parts?

In Integration by Parts, "dv" stands for the differential of the function that you will integrate, and "dx" stands for the differential of the function that you will differentiate. Generally, you want to choose "dv" to be the more complicated function, and "dx" to be the simpler function, in order to make the integration easier.

## What is the formula for Integration by Parts?

The formula for Integration by Parts is ∫ u dv = uv - ∫ v du, where "u" and "v" are the chosen functions and "du" and "dv" are their differentials.

## How many times can you use Integration by Parts on a single integral?

You can use Integration by Parts multiple times on a single integral, as long as the resulting integral is still something that can be solved. However, it is usually more efficient to choose the right combination of "u" and "v" to reduce the number of times you need to use the method.

## What are some common applications of Integration by Parts?

Integration by Parts is commonly used in various fields of science and mathematics, such as physics, engineering, and statistics. It can be used to solve integrals related to probability, moments of inertia, and heat transfer, among others.