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U-Substitution

3 min readโ€ขnovember 15, 2021

Rupi Adhikary

Rupi Adhikary


AP Calculus AB/BCย โ™พ๏ธ

279ย resources
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โšก๏ธWatch: AP Calculus AB/BC - Integration by Substitutionย ย 
U-sub, also known as integration by substitution, is one of the key components of integrals. Chances are, you've come across an integral like the one below and been completely lost on where to start.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202019-08-04%20at%203.48-kDGspmAGZPbm.png?alt=media&token=db183782-ca20-4891-9819-86806e0c3873
Fear not! In this article, we'll go over what U-Sub is and how and when you should use it! For the sake of learning the concept first, all of the examples we're going to be using will be indefinite integrals.
Let's jump right into it then!
Before using u-sub, you want to be able to write your integral as
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202019-08-04%20at%203.51-CLswyYX4NSLU.png?alt=media&token=b10a3c83-a5b1-4884-8eda-847a9a3cb5d9

Key Components

If we look at the function above, there are a couple things that we should take note of
  • f(x) - this would be represented asย cos(x)ย in the example
  • f(g(x)) - in the example, this would beย cos(x^2)
  • g(x) - the 'x' of 'f(x)', in the example at the top of this article, this would beย x^2
  • g'(x) - the derivative of g(x), in the example, this would beย 2x
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202019-08-04%20at%204.14-8g9Cp22YmPiv.png?alt=media&token=c5fa9b1f-0f9b-48a2-b7ac-f9aa834555d7

Why U-Sub?

U-substitution is all about making taking the integral of a function easier. To do this, we need to substitute a part of the function with 'u' so we can be left with something easier to work with.
We substitute g(x), with the term 'u'. This means that theย derivativeย of g(x) changes as well. G'(x) becomes the derivative of 'u' or 'du'.
This example is perfect because we can clearly see what the derivative of g(x) is but it doesn't always work out so easily. To ensure that you're correctly finding g'(x), simply take the derivative of g(x).
In the case of the example, this would play out something like this
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-wZhampMINQDk.png?alt=media&token=056e363b-664e-48ad-a9d1-1d66fe838d3c
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-4ika694qF5c8.png?alt=media&token=fa925f7d-8622-4583-adf7-b5f873608e3d
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-JoOtgl7Rdx1m.png?alt=media&token=9509afb0-9ba7-41a6-9c83-c69684fbca8d
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-CLL5V4DzMiPP.png?alt=media&token=d0398338-7f45-4ed3-84af-d31939c3d2ef
Now that we have 'u' and 'du', we can substitute them into our original integral (remember, the original integral was the integral of cos(x^2) * 2x dx).
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-NRVCIk8xiOcz.png?alt=media&token=b32c7994-08d6-42f0-b24e-ea9ebca4a5e8
Next, all we have to do is take the integral in terms of 'u', which basically just means taking the integral but instead of 'x' as you would usually put, 'u' would take its place.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-oJqWXMvkCsDU.png?alt=media&token=fe7bae21-de7b-4866-b57c-618d386f9c57
Our last step is to substitute back in g(x) for 'u'.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-Ql6HPXYDMiNt.png?alt=media&token=22521847-e4b9-4c40-9433-3d4df616b580

What if it isn't so pretty?

While this example worked out pretty well, as mentioned before, the majority of the integrals you're going to be taking won't. This means that it might take a little bit of extra work getting the integral into this type of format
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-Rv2cStKRPVwD.png?alt=media&token=c1aa300e-a30a-4b46-b1c0-57fa9b22c2ea
Say, we have the another integral like the one below
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-5KeasTcF8iod.png?alt=media&token=c6fd0776-7ed6-4a43-a39f-a9d7e7b9ec23
It looks very similar to the one that we just worked with! The only problem, is that pesky '12x'. So, how do we go about solving this?
We're going to go through the same steps that we did last time.
  • What is our g(x)? x^2
  • What is our f(x)? cos(x)
  • What is our f(g(x))? cos(x^2)
  • What is g'(x)?ย 2x
We see that everything matches upย exceptย g'(x). 12x is simply 2x multiplied by 6 right? So, if we use the rule of constants we can pull that 6 up front and be left with the 2x we need.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-QVqqQfjCcmqd.png?alt=media&token=0a115cfa-b615-43c2-abbf-2b0c4a54dddd
Now, we can take the integral as usual, but make sure not to forget about the 6 we took out!
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-R7LlvQCL968c.png?alt=media&token=fdeafca7-bbde-473e-8eca-e847ef4d0c11
And that's it! Now you know how to use u-sub like a pro! If you need some extra help and practice with this concept check out these awesome links too!

๐Ÿ“Œ Practice and help with u-sub!ย 

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