# 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

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## 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##

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