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Type 5.

Other rational expressions with the irrational function

1. If , we can use .

2. If , we can use .

3. If can be factored as , we can use .

4. If and can be factored as , we can use

Frequently occuring integrals of irrational functions are: , where and is possitive on an interval. We can exclude the case, that the polynomial has a double root.

Taking the factor (if ) or (if ) we reduce the integral to the form or , leading (by means of substitution) to the integrals:

 

or , respectively.

 

3. Integration of TRIGONOMETRIC functions

 

1. Given an integral , i.e. the integrand is a rational function in terms of and . By the substitution the integral is reduced to an integral of a rational function. If , then , , and .

2. If = , then .

3. If = - , then .

If =- , then .

4. , and even non-negative integers, then , .

5. For integrals we use following formulas:

6. , then and .

 


Date: 2015-01-02; view: 197


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Integration of simple rational fractions. Integration of rational fractions | MAIN Properties of Definite Integrals
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