Trigonometric substitution is an integration technique for expressions containing a square root of a sum or difference of squares. Learn the three standard cases (sine, tangent, and secant substitutions), how to pick the right one so a Pythagorean identity simplifies the root, and a worked example.
What trigonometric substitution is
Trigonometric substitution is an integration technique for expressions that contain a square root of a sum or difference of squares, such as √(a² − x²). By replacing x with a trig function, the square root simplifies through a Pythagorean identity and the integral becomes one you can evaluate. It extends the idea of u-substitution to a harder family of integrals.
The three cases
Which trig function you substitute depends on the form under the root.
The three trigonometric substitution cases: sine, tangent, and secant.
Each choice is built so the Pythagorean identity collapses the root: for √(a² − x²), setting x = a sinθ gives √(a² − a²sin²θ) = a cosθ.
Steps
Pick the case that matches the root, substitute x and dx, simplify using the identity, integrate in terms of θ, then convert back to x with a reference triangle. It pairs naturally with integration using trigonometric identities and, for other integrands, integration by parts.
Worked example
For ∫ dx / √(1 − x²), use x = sinθ, so dx = cosθ dθ and √(1 − x²) = cosθ. The integral becomes ∫ dθ = θ + C = arcsin(x) + C.