Hyperbolic trigonometric substitution is an integration technique that parallels ordinary trigonometric substitution: where the identity tan²+1=sec² motivates a tangent substitution for integrands involving expressions of the form y²+1, the analogous hyperbolic identity sinh²u+1=cosh²u motivates substituting y=sinh(u) for the same form, converting the integral into one involving hyperbolic functions that can be resolved via their exponential definitions. This is a topic in integral calculus concerning substitution techniques for integrals involving square roots of quadratic expressions, situated within the broader theory of hyperbolic functions (defined via exponentials) and their structural analogy to circular trigonometric functions.
Hyperbolic trigonometric substitution is an integration technique that parallels ordinary trigonometric substitution: where the identity tan²+1=sec² motivates a tangent substitution for integrands in…