Trig substitution helps you to integrate some types of challenging functions:
- Radicals of polynomial functions, like √(4 – x2),
- Rational powers of the form n/2, e.g. (x2 + 1)(3/2).
Although trig substitution is fairly straightforward, you should use it when more common integration methods (like u substitution) have failed.
The technique is very similar to u substitution: you substitute a new term (one made from integer powers of trig functions) in place of the one you have, in order to make the integration easier. At the end, you simply substitute the original function back in.
Why Is Trig Substitution a Last Resort?
Although it’s straightforward, trig substitution requires you to have a lot of background knowledge. Unlike a table of integrals, you can’t just look up an integral for a particular expression. It’s a must that you are able to recognize the trigonometric identities. Let’s look at an example to see why this is so important.
Example question: Integrate
To solve this, you need to consider all of the trig identities to see which would be a good fit. If you aren’t familiar with them, this could be a stumbling block before you’ve even started. In order to solve this particular integral, you need to recognize that it looks very similar to the trig identity
1 + tan2 x = sec2 x.
Here are the solution steps:
Step 1: : Rewrite the expression using a trig substitution (and derivative). The goal here is to get the expression into something you can simplify with a substitution:
Here, I substituted in tan2θ for x.
As the substitution for x has been made, I also had to change the “dx” to represent the derivative of tan2θ (instead of plain old derivative of “x”). So the new “dx” became sec2 θ dθ.
Step 2: : Simplify by using a trig identity. In this example, we’ve been heading towards changing 1 + tan2 x to sec2 x. There’s no magic here—if you chose the correct trig function in Step 1, you should already know which trig identity you’re going to use here:
Step 3: : Simplify using algebra (if possible). For this example, notice that we can cancel out the sec2 in the numerator and denominator,
∫ 1 dθ.
Step 4: Integrate. The integral of a constant function is just the constant * x (or constant * θ) + C, so:
∫ 1 dθ = θ + C
Step 5: Substitute your original term back in. In Step 1, I substituted tan-1 θ for x, so putting that back in gives the solution:
= tan -1 x + C
Useful Background Information
As you may be able to tell from the above example, trig substitution requires you to have some strong background skills in algebra, derivatives, and trigonometric identities.
“…any teacher of Calculus will tell you that the reason that students are not successful in Calculus is not because of the Calculus, it’s because their algebra and trigonometry skills are weak” ~ Jones (2010)
Also extremely helpful:
- Integrals of Trig Functions,
- U Substitution for Trigonometric Functions,
The following table shows how to express one of the common six trig functions as a pair of other trig functions. These may also come in handy:
Banner, A. (2007). The Calculus Lifesaver: All the Tools You Need to Excel at Calculus (Princeton Lifesaver Study Guides) Illustrated Edition. Princeton University Press.
Jones, J. (2010). Skills Needed for Success in Calculus 1.
Kouba, D. (2017). Finding Integrals Using the Method of Trigonometric Substitution. Retrieved November 9, 2020 from: https://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigsubdirectory/TrigSub.html