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.
EMBED Equation.3
.
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:
