What is trigonometric substitution and why does it work?

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Answer 1 · 2016-11-16T17:55:35

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:

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