Trig substitution - How to solve?

Trigonometric substitution (more affectionately known as trig substitution, or trig sub), is another integration method you can use to simplify integrals. It’s like u-substitution, integration by parts, or partial fractions. It takes advantage of the relationship between the sides and angles in a right triangle, allowing you to replace a more complicated value in an integral, with simpler associated values from a corresponding right triangle.

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0:31 // What is trig substitution?
1:35 // When to use trig substitution?
2:54 // What kinds of integrals use trig substitution?
7:51 // What to do when there’s no square root?
8:12 // What to do when you don’t have perfect squares?
9:45 // What do trig substitution integrals usually look like? Which trig substitution to use? (Examples of sin substitutions, tan substitutions, and sec substitutions)
15:02 // Why trig substitution works?
20:12 // How to set up for a trig substitution problem?
27:02 // How to solve trig substitution? How to do sine substitution?
27:32 // Step 1. Identify that it’s a trig sub problem
28:18 // Step 2. Decide which trig substitution to use
28:46 // Step 3. Do the setup process for trig sub
30:03 // Step 4. Make substitutions into the integral
31:18 // Step 5. Simplify the integral using whatever methods you need to, then integrate
35:04 // Step 6. Back-substitute to put the integrated value back in terms of x, instead of theta.
35:17 // How to build your reference triangle

You want to use trig sub whenever you have one of these values in your integrand: a^2-u^2, a^2+u^2, u^2+a^2, or u^2-a^2. And if you have one of those values underneath a root, or radical, that’s a dead giveaway that you might want to use trigonometric substitution.

In this video we’ll talk about how to know which trig substitution to use. Depending on the value you find in your integrand, you’ll either want to use a sine substitution, a tangent substitution, or a secant substitution. We’ll also be talking about why trigonometric substitution, and how to set up for a trig sub problem so that you can easily solve it every single time.

Here are the steps you always want to take in order to solve a trigonometric substitution problem:

1. Identify that it’s a trig sub problem. Make sure you can’t use a simpler method to solve the integral, and make sure that you have one of those a^2, u^2 values in your integrand.

2. Decide which trig substitution to use. Depending on the value in your integral, it’ll be a sine substitution, tangent substitution, or secant substitution.

3. Do the setup process for trig sub. This process is the same every time, and it’s about finding u and a, setting up your substitution values, and finding other values like what to substitute for dx.

4. Make substitutions into the integral. Make sure you replace all of the x values with a value in terms of theta, and don’t forget to replace your dx with something that includes dtheta.

5. Simplify the integral using whatever methods you need to, then integrate. You’ll definitely need at least one (probably more) trig identities in order to simplify your integral, and you may need to use some other integration methods too. Don’t be afraid to keep going, these problems can get a little long sometimes. Once you can solve the integral, go ahead and integrate.

6. Back-substitute to put the integrated value back in terms of x, instead of theta. For this step you’ll need to draw your reference triangle, and refer to it to replace any trig functions that are in terms of theta with values that are in terms of x. Don’t forget to add C if it’s an indefinite integral, and this is your final answer! :)

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[JerrodEstell]

YouTube is a gold mine. This education cannot be priced.

[dugunu]

I want to thank professor Krista King for her amazing explanation of how to solve this kind of problem. Without your help, it could be impossible for me to understand. thanks so much

[marvhartigan]

you really live up to your name Krista, YOU ARE REALLY A KING, KING OF

[Sequence]

World Class explanation, thank you so much for this video. Many educators (won’t mention names because they have a lot of views and their fan base might kill me) skip through the nuances that make trig substitution easier to understand. I know you put in a lot of work I can tell just by the quality, you make no assumptions at all and go through even the nitty gritty stuff. Your explanation is “King”

[mayeboy]

Boom, boom I'm back again. Trigging and a subbing with Krista my friend!

[John-fzli]

Thank you, Krista, before your video, I've watched 4 different videos about trig substitution (1- My teacher's video, 2- Khan Academy, 3- Organic chemistry, 4- Nancy pi) Your video is most helpful, thank you for your great help.

[end]

It's crazy to think how smart I might have been if I grew up in today's age! What a blessing it is to be able to jump right onto the internet, on YouTube, and find someone explaining things you're studying better than your own teacher did in school! WONDERFUL !!!

[lyrafiddle]

This is so good -- I haven't looked at this material since I was in high school in 1993, and now I'm trying to teach myself about differential forms and geometric algebra, and realising that I have forgotten some of these basic techniques from calculus! You present these topics in a way that is well-paced and methodical without oversimplification. Congratulations!

[georgepage]

The only way to learn is adequate information and she did!
Thank's to you Ms K².

[thevan]

You are using a very high quality microphone here, such crispy clean audio. Great choice thank you.

[mathsciencenerdstation]

Thank you so much!!! This video helped me more than words can describe! Love the way you break down concepts, when and how to use rules, and your thinking of why you are doing what you are doing along the way is just AWESOME! I went from an understanding of 3-4 to 9-10 on trig sub! Can't thank you enough! Thank you for your time doing these videos!

[ThatDrummerFrank]

Your skill at presenting material is out of this world. I struggled with this material so much last semester. Doing a review for multi variable currently and happened across your videos. Thanks for doing what you do.

[safouenmissaoui]

Incroyable, je n'est jamais cru que c'est tellement facile à résoudre

[jacksonfev]

Thanks for not doing some cheesy problem and instead actually showing a good example of what you are trying to explain.

[federicoguglielmotti]

I'm trying to self-study the calculus AB course and I was struggling to understand trig sub, but this video was a literal godsend - thank you so much for this! It made everything seem really approachable

[livdennen]

Thank you SO MUCH!! I was so lost before this video. I appreciate all the work you put into this!

[GouravEditZ-_-]

thankyou very much for this amazing lecture, fantastic explaination .love from India.🙂

[catholic_zoomer_bro]

This video is much better than the previous, good job.

[mandlangonyama]

You really know how to explain math while keeping it simple and fun...thank you

[lebulle]

Thanks. No one ever explains where it comes from, they just go straight to the method