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Integration

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:

right angle triangle with angle half x, opposite side u, adjacent side 1 and hypotenuse square root of (1 plus u squared)

We deduce cos (half x) = 1 over square root of (1 plus u squared) 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.

Example

Consider .

Let u = tan then du = half (1 + u squared) dx and .

The integral simplifies to :

tan (half x) + c

Exercise

Try all 5 of these.

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