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๐Ÿ‘‘ย  Unit 1: Limits & Continuity

๐Ÿถย  Unit 8: Applications of Integration

โ™พย  Unit 10: Infinite Sequences and Series (BC Only)

6.11 Integrating Using Integration by Parts

#integration

#byparts

โฑ๏ธย ย 3 min read

written by

Catherine Liu

catherine liu


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AP Description & Expectations ๐ŸŽฏ

  • Enduring Understanding FUN-6: Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.ย 

  • 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 functionโ€™s derivative and the others antiderivative. Although this sounds confusing at first, we can build this concept from the product rule:ย ๐Ÿฅ‡

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Whenever thereโ€™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 gโ€™(x). If other methods like u-substitution donโ€™t work, try integration by parts. ๐Ÿ’ฏ

Another Form of the Equation โ“

Sometimes, you might see the equation written like this:

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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 = fโ€™(x)dx. Same thing for V: If V = g(x),ย  then dV = gโ€™(x)dx. This new way of writing the integration by parts formula is just a sneakier version of the original formula. ๐Ÿคญ

How and Why Would I Use This Formula? ๐Ÿค”

The best way to explain this formula is through an example. Take the following integral:

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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. Letโ€™s recall the formula:

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The first step should be to figure out what f(x) and gโ€™(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.ย 

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 โ€œEโ€ side should be gโ€™(x) and the function closer to the โ€œLโ€ side should be f(x).ย 

For our example problem, that means:

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We know we need g(x) and fโ€™(x) to use the integration by parts formula, so letโ€™s find those now:

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Steps to Take for Integration by Parts

  1. Check to make sure that u-substitution and other methods donโ€™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 gโ€™(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.ย 

Worked Examples ๐Ÿ“”

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