This concept covers two techniques for integrating rational functions and rational functions involving square roots of quadratics: partial fraction decomposition, which rewrites a rational function with a factorable denominator as a sum of simpler fractions integrable via substitution or logarithmic antiderivatives, and trigonometric substitution combined with completing the square, which converts an integrand containing √(1 − u²)-type expressions into a purely trigonometric integral. It relies on the formal conditions for partial fractions (numerator degree strictly less than denominator degree), the Pythagorean trigonometric identity (1 − cos²θ = sin²θ), and the known antiderivative of secant. This belongs to single-variable integral calculus, within the theory of integration techniques for rational and algebraic functions, extending prior individual techniques (substitution, logarithmic integration, trigonometric substitution) into a combined decision framework.
This concept covers two techniques for integrating rational functions and rational functions involving square roots of quadratics: partial fraction decomposition, which rewrites a rational function w…