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.
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.
In this video you'll learn:
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
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! :)