Trigonometric substitutions

Integrals involving

Expression	Substitution	Interval
$\sqrt{a^{2} - x^{2}}$	$x = sin θ$	$θ \in [- \frac{π}{2}, \frac{π}{2}]$
$\sqrt{a^{2} + x^{2}}$	$x = tan θ$	$θ \in (- \frac{π}{2}, \frac{π}{2})$
$\sqrt{x^{2} - a^{2}}$	$x = \frac{1}{cos θ}$	$θ \in [0, \frac{π}{2}) \cup [π, \frac{3 π}{2})$

Geometry

\sqrt{a^{2} - x^{2}}

Geometry

\sqrt{a^{2} + x^{2}}

Geometry

\sqrt{x^{2} - a^{2}}