Question

What is trigonometric substitution and why does it work?

Answer 1

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`