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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. 
, ò and ï – even non-negative integers, then 
, 
.
5. For integrals 
we use following formulas:
6. 
, then 
and 
.
Date: 2015-01-02; view: 1384
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