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# 6.11 Integrating Using Integration by Parts

Catherine Liu

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## AP Description & Expectations π―

• Enduring Understanding FUN-6: Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.Β
• Essential Knowledge FUN-6.E.1: Integration by parts is a technique for finding antiderivatives.

## What is Integration by Parts?

Integration by parts is a method of integration that transforms an integral of a product of functions into an integral of the product of one functionβs derivative and the others antiderivative. Although this sounds confusing at first, we can build this concept from the product rule:Β π₯
Whenever thereβs an integral where the integrand (the thing being integrated) is a product of two functions, one of the functions will be f(x) and the other will be gβ(x). If other methods like u-substitution donβt work, try integration by parts. π―

## Another Form of the Equation β

Sometimes, you might see the equation written like this:
This is the same concept in a different form.
One of the functions will be U and the other will be V. However, we still need to know what dV and dU actually mean. If U is a function of x, then U = f(x). Taking the derivative of both sides, we find that dU = fβ(x)dx. Same thing for V: If V = g(x),Β  then dV = gβ(x)dx. This new way of writing the integration by parts formula is just a sneakier version of the original formula. π€­

## How and Why Would I Use This Formula? π€

The best way to explain this formula is through an example. Take the following integral:
At first glance, you may try u-substitution, but this method will take you nowhere. Instead, notice that the integrand is a product of two functions. This is an opportunity to use integration by parts. Letβs recall the formula:
The first step should be to figure out what f(x) and gβ(x) should be. This step is probably the most challenging of this method, because you have to choose which is which in order to make the problem work.Β
This is where a handy acronym comes into play: L.I.A.T.E. It stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential.
The function closer to the βEβ side should be gβ(x) and the function closer to the βLβ side should be f(x).Β
For our example problem, that means:
We know we need g(x) and fβ(x) to use the integration by parts formula, so letβs find those now:

## Steps to Take for Integration by Parts

1. Check to make sure that u-substitution and other methods donβt work.
2. Check to make sure that the integrand is a product of two functions.
3. Use L.I.A.T.E. to pick which function should be f(x) and which should be gβ(x).
4. Plug into the formula and integrate.
Note: You may have to use integration by parts more than once if your resulting integral is also a product of two functions that can't be solved using u-substitution.Β

## Worked Examples π

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