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# 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

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