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## 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.

### Example

Consider .

Let u = tan then and .

The integral simplifies to : ### Exercise

Try all 5 of these. dx = Working:

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