Trigonometric substitution is a calculus technique for integrals that contain square roots such as sqrt(a^2 - x^2), sqrt(a^2 + x^2), or sqrt(x^2 - a^2). These forms are difficult because algebraic substitution often leaves the square root just as complicated. Trig substitution works because trigonometric identities can turn these square roots into simpler expressions.
It matters because it connects integration, geometry, and right triangle reasoning in one method.
The main idea is to match the square-root form to a Pythagorean identity, choose a trigonometric substitution for x, and rewrite dx. After integrating in terms of the angle, you convert back to x using a right triangle or an inverse trig relationship. For example, sqrt(a^2 - x^2) matches 1 - sin^2(theta) = cos^2(theta), so x = a sin(theta).
The final answer should usually be written in terms of the original variable unless the problem asks otherwise.
Key Facts
- For sqrt(a^2 - x^2), use x = a sin(theta), dx = a cos(theta) dtheta, and sqrt(a^2 - x^2) = a cos(theta).
- For sqrt(a^2 + x^2), use x = a tan(theta), dx = a sec^2(theta) dtheta, and sqrt(a^2 + x^2) = a sec(theta).
- For sqrt(x^2 - a^2), use x = a sec(theta), dx = a sec(theta)tan(theta) dtheta, and sqrt(x^2 - a^2) = a tan(theta).
- The key identities are 1 - sin^2(theta) = cos^2(theta), 1 + tan^2(theta) = sec^2(theta), and sec^2(theta) - 1 = tan^2(theta).
- A right triangle helps convert back: if x = a sin(theta), then sin(theta) = x/a and the missing side is sqrt(a^2 - x^2).
- A typical result may include inverse trig terms, such as theta = arcsin(x/a), theta = arctan(x/a), or theta = arcsec(x/a).
Vocabulary
- Trigonometric substitution
- A method for evaluating certain integrals by replacing a variable with a trigonometric expression that simplifies a radical.
- Pythagorean identity
- An equation relating trigonometric functions, such as sin^2(theta) + cos^2(theta) = 1, that comes from the Pythagorean theorem.
- Back-substitution
- The process of rewriting the final answer from the angle variable back into the original variable.
- Right triangle diagram
- A triangle used to express trigonometric functions of theta in terms of x and constants after a substitution.
- Radical expression
- An expression containing a root symbol, such as sqrt(a^2 - x^2), that often motivates trig substitution.
Common Mistakes to Avoid
- Choosing the wrong substitution for the square-root form. The form sqrt(a^2 + x^2) should match x = a tan(theta), not x = a sin(theta), because it uses 1 + tan^2(theta) = sec^2(theta).
- Forgetting to replace dx. Every substitution changes both x and dx, so leaving dx unchanged makes the transformed integral incorrect.
- Failing to convert the answer back to x. An indefinite integral that began with x should usually end with x, not only theta.
- Dropping absolute value or sign restrictions without checking the interval. Expressions like sqrt(a^2 sec^2(theta)) require attention to whether sec(theta) is positive on the chosen theta interval.
Practice Questions
- 1 Evaluate the integral ∫ dx / sqrt(9 - x^2) using the substitution x = 3 sin(theta).
- 2 Evaluate the integral ∫ sqrt(x^2 + 16) / x^2 dx using the substitution x = 4 tan(theta).
- 3 Explain why sqrt(x^2 - 25) suggests the substitution x = 5 sec(theta), and describe the right triangle you would use to convert back to x.