Trigonometric Substitutions

Expression	Substitution	Identity Used	Result
sqrt( $a^2$ - $x^2$ )	x = a sin(theta)	1- $sin^2$ = $cos^2$	sqrt becomes a*cos(theta)
sqrt( $a^2$ + $x^2$ )	x = a tan(theta)	1+ $tan^2$ = $sec^2$	sqrt becomes a*sec(theta)
sqrt( $x^2$ - $a^2$ )	x = a sec(theta)	$sec^{2-1}$ = $tan^2$	sqrt becomes a*tan(theta)

After substitution: Express everything in terms of theta, integrate, then convert back using a right triangle.

Common pitfall: Forgetting to change dx. If x = a sin(theta), then dx = a cos(theta) d(theta).

Example: integral dx/sqrt( $x^{2+4}$ ). Let x = 2tan(theta). dx = 2 $sec^2$ (theta)d(theta). sqrt( $x^{2+4}$ ) = 2sec(theta). Integral = integral sec(theta)d(theta) = ln|sec(theta)+tan(theta)| = ln|x/2 + sqrt $\frac{x^2+4}{2}$ | = ln|x+sqrt( $x^{2+4}$ )| - ln2.