What is trigonometric substitution and why does it work?

Trig substitution is an integration substitution involving a trig function. It used to solve problem such as

## int sqrt(a^2+-x^2) dx ##, and ## int sqrt(x^2+-1^2) dx ## ## int 1/sqrt(a^2+-x^2) dx ##, and ## int 1/sqrt(x^2+-1^2) dx ##

and various other similar forms. They work simply because of the various trig identities

Example:

## int 1/sqrt(1-x^2)dx##

Let ##x=sinu => dx/du=cosu##, Hence ##int ...dx=int ..cosudu##

Using the trig identity ##sin^2A+cos^2A-=1## we have

## sin^2u+cos^2u = 1 ## ## :. cos^2u = 1-sin^2u ## ## :. cos^2u = 1 - x^2## ## :. cosu = sqrt(1 - x^2)##

Substituting into the integral we have:

## int 1/sqrt(1-x^2)dx = int 1/cosu*cosdu## ## :. int 1/sqrt(1-x^2)dx = int du## ## :. int 1/sqrt(1-x^2)dx = u + C## ## :. int 1/sqrt(1-x^2)dx = arcsinx + C##