MATH1231 - Trigonometric and Hyperbolic Substitutions

The aim of trig and hyperbolic substitutions is to provide the easiest substitution for some common integrals. There’s nothing fancy, just a normal substitution (the end working out is often done by drawing a triangle to find the various cos, sin, tan, of θ)

E.g. integrals of the form $\surd(\pm x^2 \pm a^2)$

(1)
\begin{align} \begin {array} {c c c} Integrand Expression & Trigonometric Substitution & Hyperbolic Substitution \\ \surd(a^2 - x^2) & x = asin(θ) & x = atanh(θ) \\ \surd(a^2 + x^2) & x = atan(θ) & x = asinh(θ) \\ \surd(x^2 - a^2) & x = asec(θ) & x = acosh(θ) \\ \end {array} \end{align}

We usually use trig substitutions for efficiency and ease of use.