Understanding Trigonometric Substitution

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  • 0:10 Trig Substitution
  • 1:13 Example #1
  • 2:24 Trig Substitution Cheat Sheet
  • 4:04 Example #1, Continued
  • 6:58 Example #2
  • 9:29 Lesson Summary
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Lesson Transcript
Instructor: Zach Pino
Sometimes a simple substitution can make life a lot easier. Imagine how nice it would be if you could replace your federal tax form with a 'Hello, my name is...' name badge! In this lesson, we review how you can use trigonometry to make substitutions to simplify integrals.

Trig Substitution

When I was learning how to drive, I always asked my mom if I could drive to the mall. She said I was not allowed to drive or get my driver's license until I learned how to drive a stick shift. Now I didn't think this was particularly fair, because my goal was simply to get to the mall and I wanted to do this by driving a car. I knew that a stick shift was the hard way to go; instead, I wanted to take my stick shift car and turn it into an automatic because I knew that driving an automatic would be a lot easier.

This is a lot like a trig substitution in math. The goal with trig substitution is to use substitution based on trig identities. We're going to use substitution based on right triangles to make integration easier. So here, your goal might be to evaluate an integral, but you want to do that by finding an anti-derivative. Before you use the right substitution, you might have a complicated mess on your hands, but after using trig substitution, life might be a little simpler.

Example #1

So let's take an example. Let's say you have the integral of (1/ the square root of (1 - x^2))dx. I can map this to a right triangle, that is, one where I've got a, b and c, where c is the hypotenuse. By the Pythagorean theorem, I know that a^2 + b^2 = c^2. One right triangle that looks a little bit like this equation is if I have 1 on the hypotenuse, x on one of the sides and then by the Pythagorean theorem, I know that the other side is the square root of (1 - x^2).

So why do I map it like this? Well, I want to keep this triangle as simple as possible but still make it look like my integral. So here I've got a 1 (that's for this value here), and I've got an x for this value here and on this third side, as a result of the other two, I have the square root of (1 - x^2). I draw out my triangle like this so that this complicated-looking term is just one side of my triangle.

List of trig substitutions
Trig Substitutions Cheat Sheet

Trig Substitution Cheat Sheet

I'll be honest. In general, trig substitutions are very difficult. It's hard to see them, and usually when you see a trig substitution, you might want to look at doing your problem differently. But sometimes you can't avoid them. So what are some rules that might help us find substitutions that make a little bit of sense? Well, if you have a function that depends on some constant squared, C^2 + x^2, then you should consider using the substitution x equals C times the tangent of theta, which written in symbol form is x=C * tan(theta). Here, you're going to replace x in your integrand with a new variable, theta. When you use this particular substitution, keep in mind that 1 plus the tangent squared of theta is equal to the secant squared of theta (1 + tan^2(theta) = sec^2(theta)). This is a trig identity.

If on the other hand, you have a function that depends on some constant squared minus x^2, you might want to consider the substitution x equals C times the sine of theta (x=C * sin(theta)). When you use this substitution, you might want to remember the trig identity 1 minus the sine squared of theta equals the cosine squared of theta (1 - sin^2(theta) = cos^2(theta)).

Using trig substitution for example #1
Example 1 Trig Substitution

Finally, if you have x^2 - C^2 - so some constant squared - you might want to consider using the substitution x equals C times the secant of theta (x=C * sec(theta)). If you use this trig substitution, keep in mind the trig identity secant squared of theta minus 1 equals the tangent squared of theta (sec^2(theta) - 1 = tan^2(theta)).

Example #1, Continued

Let's go back to our example of (1 / the square root of (1 - x^2))dx. Because we have 1 - x^2, that's like 1^2 - x^2. So according to our little cheat sheet of substitutions, we might want to consider using the substitution x=C * sin(theta), where C is just 1.

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