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
, ( 4 )
where a function of the argument x
is a rational one■
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
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
Ex. 10. Evaluate the indefinite integral .
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