Calculus › Section I · Integration techniques and improper integrals › Topic I.4
Trigonometric substitution
Square roots like √(a² − x²) appear in circle areas, arc lengths and distance calculations in physics — and they resist substitution, parts and every algebraic trick. Trigonometric substitution makes the root vanish: replace x with a trigonometric function chosen so that a Pythagorean identity turns the expression under the root into a perfect square.
In this topic
- The three cases
- Choosing θ so the root stays positive
- How a substitution proceeds
- Worked example: the semicircle root
- The reference triangle
- Worked example: a plus-sign root
- Worked example: the reversed root
- Applied: the field above a charged rod
- Common mistakes to avoid
- Key takeaways
This topic is part of Wuanberri Pro. Checking your access…
Standards: Beyond AP