Equations · Calculus and analysis
Integration by parts
The product rule turned around: trade one integral for another that is easier to compute.
∫ u dv = uv − ∫ v du
Aethel says
the integral of u d v equals u v minus the integral of v d u
What it says
Split what is being integrated into two parts, u and dv. The integral equals u times v, minus a new integral in which the roles are swapped.
A good choice of u is one that gets simpler when differentiated, such as a power of x multiplied by an exponential.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| ∫ | the integral of | an indefinite integral |
| u, v | u, v | two functions of the same variable |
| du, dv | d u, d v | their differentials: the derivative times d x |
More on∫Integral
Where you meet it
Calculus courses, Fourier analysis and the proofs of many series.
Questions
How do you say Integration by parts out loud?
∫ u dv = uv − ∫ v du is read: “the integral of u d v equals u v minus the integral of v d u.” That is exactly what Aethel says when it meets the formula in a book.
What does ∫ mean in Integration by parts?
∫ (said “the integral of”) is an indefinite integral.
What does u, v mean in Integration by parts?
u, v (said “u, v”) is two functions of the same variable.
Can I listen to the equations in my own textbook like this?
Yes. Upload a PDF, EPUB or text file and Aethel reads it aloud with every equation spoken as maths, the way it reads this one, and highlights each sentence as it goes. You can start free.