Choosing the right trigonometric substitution
Match a quadratic radical to a Pythagorean identity so the square root simplifies instead of becoming worse.
LaTeX article Updated July 11, 2026
The three structural patterns
For a² − x², sine creates 1 − sin²θ = cos²θ. For a² + x², tangent creates 1 + tan²θ = sec²θ. For x² − a², secant creates sec²θ − 1 = tan²θ.
Complete the square first when the quadratic is shifted or scaled.
Track the differential and domain
Every substitution changes dx as well as the radical. Choose a θ-interval that makes the square-root simplification consistent with a nonnegative radical.
A reference triangle gives a reliable back-substitution without memorizing inverse identities.
Know when a simpler substitution exists
A radical alone does not force trig substitution. If the derivative of the expression under the root is already present, ordinary u-substitution is shorter and clearer.
Worked example
Common mistakes
- Choosing tangent for the a² − x² pattern.
- Forgetting to transform dx.
- Back-substituting without a triangle or a valid inverse relation.
Keep these ideas
- Match the sign pattern to a Pythagorean identity.
- Complete the square before classifying.
- Check whether u-substitution is simpler first.