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Trigonometric substitution

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Trigonometric Substitution

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 A reference table. When the integrand contains the square root of a squared minus x squared, substitute x = a sin theta. For the square root of a squared plus x squared, substitute x = a tan theta. For the square root of x squared minus a squared, substitute x = a sec theta. Expression containsSubstitute √(a² − x²)x = a sinθ √(a² + x²)x = a tanθ √(x² − a²)x = a secθ
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.

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