Integration by Trigonometric Substitution 2
We assume that you are familiar with the material in integration by trigonometric substitution 1 and geometric representation of trigonometric functions.
Various methods are available for evaluating the integrals of rational functions (ratios of two polynomials). It has been discovered that a fractional expression involving trig functions can always be transformed into a rational function.
Let u = tan . We can represent this in the following diagram:
We deduce and . Hence,
Since all other trig functions can be written in terms of sin x and cos x all fractional expressions of trig functions can always be transformed into a rational function.
Let u = tan then and .
The integral simplifies to :
Try all 5 of these.