This concept combines two techniques for integrating rational functions with an irreducible quadratic denominator: completing the square to rewrite a quadratic as a difference (or sum) of squares, followed by trigonometric substitution to convert the resulting integral into one expressible purely in trigonometric functions. It relies on the algebraic identity for completing the square, the substitution rule u = a·sec(θ) for denominators of the form u² − a², and the Pythagorean trigonometric identity 1 + tan²θ = sec²θ. This belongs to single-variable integral calculus, specifically the theory of integration techniques for rational functions whose denominators cannot be factored over the rationals, situating this method as an alternative to partial fractions when the quadratic is irreducible.
This concept combines two techniques for integrating rational functions with an irreducible quadratic denominator: completing the square to rewrite a quadratic as a difference (or sum) of squares, fo…