Examples of integrals found using hyperbolic substitutions

Question

I've read that various types of integrals usually determined by involving $\tan$ and $\sec$ into the mix can sometimes be found more easily using hyperbolic functions.

As I'm not very familiar with the latter family, could someone give me some examples?

Very simple example: $$\int \frac 1 {1+x^2}\,\mathrm dx.$$

Answer 1

Very simple example: $$\int \frac 1 {1+x^2}\,\mathrm dx.$$

Substituting in $x=\sinh\theta$ results in $\int\mathrm{sech}\,\theta\,\mathrm d\theta,$ which can be derived using trickery or using the substitution $u=e^\theta.$ But since this is a standard integral and subtituting in $x=\tan\theta$ much quicklier gives the answer $\arctan(x),$ this is is a non-example of hyperbolic substitution being easier than trigonometric substitution!

Otherwise, whenever the substitution $x=\tan\theta$ or $x=\sec\theta$ is called for, $x=\sinh\theta$ or $x=\pm\cosh\theta,$ respectively, are invariably preferable, due to nicer principal domains and being easier to differentiate and integrate. Here are some examples:

	integrand	substitution	result
1	$\displaystyle\frac{x^2}{\sqrt{x^2+4}}$	$x=2\sinh\theta\quad\color{green}✔$	$4\int\sinh^2\theta\,\mathrm d\theta$
	$\displaystyle\frac{x^2}{\sqrt{x^2+4}}$	$x=2\tan\theta\quad\color{red}✗$	$4\int\tan^2\theta\sec\theta\,\mathrm d\theta$
2	$\displaystyle\frac{\sqrt{x^2-4}}{x^2}\quad(x\leq-2)$	$x=-2\cosh\theta\quad\color{green}✔$	$-\int\tanh^2\theta\,\mathrm d\theta$
	$\displaystyle\frac{\sqrt{x^2-4}}{x^2}\quad(x\leq-2)$	$x=2\sec\theta\quad\color{red}✗$	$-\int\tan^2\theta\cos\theta\,\mathrm d\theta$
3	$\sqrt{9-x^2}$	$x=3\sin\theta\quad\color{green}✔$	$9\int\cos^2\theta\,\mathrm d\theta$
	$\sqrt{9-x^2}$	$x=3\tanh\theta\quad\color{red}✗$	$9\int\mathrm{sech^3}\,\theta\,\mathrm d\theta$