Hi everyone! We’ve already seen this integration technique in Lesson 6 (link here). Today we’ll add another layer so that we can apply this technique in more examples.
Lesson 7: Trigonometric Substitution (part 2)
- Solve integration problems involving a quadratic polynomial or the square root of a quadratic polynomial.
- Volume 2, Section 3.3 Trigonometric Substitution (link to textbook section)
- Integration – Trigonometric Substitution
How can we use trigonometric substitution to evaluate integrals like
Algebra refresher: completing the square
This algebra technique is the only new thing we’re introducing in this lesson. If you need a refresher on completing the square, you can watch the following videos.
- The video linked here introduces the concept of completing the square.
- The vides linked here, here, and here show a handful of examples of different types.
Don’t forget: if you’re completing the square for a quadratic expression where the coefficient on the quadratic term is not , you’ll have to factor it out from the quadratic and linear terms first and then complete the square inside the parentheses:
Trigonometric substitution with linear terms–examples
Now we’re ready to get back to evaluating integrals. In all of these examples, the goal is to apply trigonometric substitution, but to know which substitution to make, we must recognize the relevant factors as sums or differences of squares , , or .
The technique of trigonometric substitution can be applied to some integrals even when they don’t appear to fit the structure of Lesson 6 (link here). The integrand can be prepared for trig substitution by completing the square and potentially forming another (non-trigonometric) substitution first. Sometimes after performing a trig substitution we’ll be left with an integral we can evaluate directly but sometimes we’ll have to apply another technique from our toolbox.