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Дисциплины:






A) Application of power reduction formulas



, ( 11 )

Ex. 14. .

Ex. 15.

.

Ex. 16.

.

 

B) Application of product formulas

1) ; 2) ;
3) ( 12 )

Ex. 17.

.

POINT 3. IRRATIONAL FUNCTIONS

Linear and linear-fractional irrationalities

 

Evaluation of indefinite integrals of the type

, ( 13 )

with so-called linear irrationality , reduces to integration of a rational function of one variable t with the help of the substitution

( 14 )

■From (14) we get

We’ve got an integral of a rational function .■

Indefinite integrals of the type

( 15 )

with linear-fractional [homographic] irrationality is reduced to that of rational function by the substitution

( 16 )

Prove this assertion yourselves.

Ex. 18.

.

Ex. 19.

Quadratic irrationalities. Trigonometric substitutions

Indefinite integral of the type

( 17 )

reduces to that with an integrand, depending on , by a trigonometric substitution

. ( 18 )

The same is true for an integral

( 19 )

if one introduces a substitution

, ( 20 )

and for an integral

( 21 )

provided a substitution

( 22 )

■Let’s consider the integral (17) and put . We’ll have ,

,

where .■

Consider the integrals (19), (21) yourselves.

Ex. 20.

.

Ex. 21.

.

Ex. 22.

Quadratic irrationalities (general case)

Indefinite integral of the form

( 23 )

can be reduced to one of integrals (17), (19), (21) with the help of the substitution

( 24 )

There are many other methods of evaluating the integral (23). For example one can reduce integration to that of rational function with the help of Euler substitutions.

The first Euler substitution (if ):

; ( 25 )

the second Euler substitution (if ):

; ( 26 )

the third Euler substitution (if the trinomial has two real roots ):

. ( 27 )

Ex. 23.

.

General note

There are inexpressible integrals which can’t be expressed in terms of elementary functions.

Examples:





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