Point 2. Trigonometric functions
In this point we study the 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 t
is a rational one■
Ex. 6.
Ex. 7. Direct evaluation
of the integral
with the help of UTS of
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
)
Ex. 9. Calculate the indefinite integral
.
The
integrand
is odd function with respect to
,
because of
,
and so
.
Ex. 10.
.
Ex. 11. Find the
indefinite integral
.
Ex. 12. Calculate the
indefinite integral
.
The integrand
is even function with respect both to
and
.
So
Note 1. Substitutions of this point can be applicable to some irrational functions of and .
Ex. 13. Evaluate the indefinite integral
.
Ex. 14. For positive
.
Some other methods
a) Application of power reduction formulas
,
( 11 )
Ex. 15.
.
Ex. 16.
.
Ex. 17.
.
b) Application of product formulas
1)
;
2)
;
3)
( 12 )
Ex. 18.
.