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?
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.
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.
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.
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.
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.
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.