Trigonometric substitution

Trigonometric substitution is an integration technique that replaces an algebraic expression with a trigonometric parametrization. It uses trigonometric identities to simplify radicals and related expressions.

Pythagorean trigonometric identity: The identity sin²θ + cos²θ = 1, together with equivalent forms involving tangent and secant. The identities make radicals such as √(1 − sin²θ) simplify to trigonometric functions.
Integration: The mathematical process of finding antiderivatives or accumulating quantities over intervals. Trigonometric substitution is one technique within integration.
Integral of √(a² − x²): An antiderivative problem involving the square root of a positive constant squared minus the variable squared. The substitution x = a sin θ removes the radical through the Pythagorean identity.
Hyperbolic substitution: A substitution using hyperbolic functions to simplify expressions involving differences or sums of squares. It uses identities such as cosh²u − sinh²u = 1 where trigonometric substitutions use circular identities.
Change of variables: A method that transforms an integral by replacing its variable and accounting for the derivative of the transformation. Trigonometric substitution is a specialized change of variables with an identity-guided choice.
Radical expression: An algebraic expression containing a root, such as a square root. Square-root expressions involving quadratic polynomials are the standard targets of the method.
Integral of √(x² − a²): An antiderivative problem involving the square root of the variable squared minus a positive constant squared. The substitution x = a sec θ simplifies the radical using sec²θ − 1 = tan²θ.
Euler substitution: A rational substitution method for integrals involving a linear expression and the square root of a quadratic. It can rationalize the same quadratic radicals without introducing trigonometric functions.
Inverse trigonometric functions: Functions that return an angle from a trigonometric ratio, including arcsine, arccosine, and arctangent. They often express the final answer after the substitution is reversed.
Quadratic polynomial: A polynomial whose highest-degree term has degree two. Completing the square in a quadratic reveals the forms matched to sine, cosine, or tangent.

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