Trigonometric Substitution - Calculus 2

In the previous lesson we focused on ways to solve trigonometric integrals that involved either sine and cosine, or secant and tangent. Now we can use those skills to help us with another advanced integration technique known as trigonometric substitution.

Often feared by many students due to its initial complexity upon introduction, trig substitution is an integration technique that allows us to integrate functions that include certain radical expressions. However, despite what you might have heard about trig substitution (or maybe not have heard), it is not as confusing or difficult as its reputation would have you believe.

The key to using trig substitution successfully is being able to correctly identify which trig function you need to substitute into an integral (sine, secant, or tangent), and having a solid understanding of how trig functions relate the sides of a triangle. The first part may seem obvious given the topic at hand, but perhaps the triangle part seems a bit out of place.

The reason we need to understand how trig functions relate the sides of a triangle is due to the fact that trig substitution involves a change of variables for the integral, similar to u-substitution from earlier in calculus. However, instead of rewriting the integral in terms of u, with trig substitution, we rewrite the integral in terms of a different variable, θ, the greek letter "theta."

This variable, θ, represents the angle of a right triangle (an angle other than the right angle) that is commonly used with trigonometric functions. When we integrate our rewritten integral, our answer will be in terms of θ, but our original integral was in terms of x, and so we cannot leave our answer in terms of θ. Instead, we need to rewrite the answer and substitute back into terms of x by using our original substitution and a triangle. This is usually where trig substitution earns its reputation for being "annoying" or "difficult" because setting up the triangle can be tricky if your trigonometry skills are a bit rusty. Thankfully, everything you need to know is covered in this new lesson, so let's get to it!

In this lesson, you will learn:

• How to choose a trig function (sinθ, secθ, or tanθ) to substitute into an integral based on a given radical expression
• How to use that substitution to rewrite an integral from in terms of x to be in terms of θ (theta) and then solve it
• How to switch a solution in terms of θ back into terms of x to complete the trig substitution process

I hope you find this video to be helpful!

-Josh