Главная Обратная связь

Дисциплины:






POINT 2. TRIGONOMETRIC FUNCTIONS



 

In this point we study methods of integration of a rational function

( 2 )

or two arguments , .

Universal trigonometrical substitution

 

Theorem 3.Integration of a function (2) always reduces to that of a rational function of one variable t with the help of so-called universal trigonometrical substitution (UTS)

( 3 )

■On the base of (3) we have

,

,

.

Therefore,

, ( 4 )

and

,

where a function of the argument x

is a rational one■

Ex. 6.

Ex. 7. Direct evaluation of the integral with the help of UTSof the form leads to complicated integral (verify!). We’ll reduce it to the integral of preceding example by changing a variable, namely

.

Ex. 8.

Other substitutions

 

I. If a function (2) is odd with respect to ,

( 5 )

then it can be transformed to the next form:

,

where is a rational function of one variable . Substitution

( 6 )

reduces integration of the given function to that of a rational function of t.

II. If a function (2) is odd with respect to ,

, ( 7 )

one can bring it to the form

( is a rational function of ) and apply the substitution

( 8 )

III. If a function (2) is even with respect both to and

( 9 )

it’s transformable into a rational function of ,

,

and can be integrated with the help of one of substitutions

( 10 )

Note 1.Substitutions of this point can be applicable to some irrational functions of and .

Ex. 9. Calculate the indefinite integral .

The integrand is odd function with respect to , because of

,

and so

.

Ex. 10. Evaluate the indefinite integral .

Ex. 11.

.

Ex. 12. Find the indefinite integral .

Ex. 13. Calculate the indefinite integral .

The integrand is even function with respect both to and . So

Some other methods





sdamzavas.net - 2020 год. Все права принадлежат их авторам! В случае нарушение авторского права, обращайтесь по форме обратной связи...