What is trigonometric substitution and why does it work?

1 Answer
Nov 16, 2016

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#