2. Main methods of integration
Integration by substitution ( or change of variable )
Steps for Integrating by Substitution—Indefinite Integrals:
1. Choose a substitution u = g(x), such as the inner part of a composite function.
2. Compute
.
3. Re-write the integral in terms of u and du.
4. Find the resulting integral in terms of u.
5. Substitute g(x) back in for u, yielding a function in terms of x only.
6. Check by differentiating.
If f(x) is continuous function, F(x)- its antiderivative and φ(х)- differentiable function, then
In
the particular case
Example
4. To
find
. Notice that the numerator is the derivative of the denominator
Let
. Differentiating gives
and hence
.
Substituting this change of variable the integral becomes
Now by expressing this result in terms of we have shown that
.
Integration by parts
By
the Product Rule for Derivatives,
.
Thus,
.
This formula for integration
by parts often makes it possible to reduce a complicated integral involving a product to
a simpler integral. By letting
we
get the more common formula for integration by parts:
.
Example
5. Find
.
Let
and
and
.
Thus,
.
It is possible that when you set up an integral using integration by parts, the resulting
integral will be more complicated than the original integral. In this case, change your
substitutions for u and dv.
3. Integration of fractional rational functions
Selecting the proper rational fraction
Suppose
is a rational function; that is,
and
are
polynomial functions. If the degree of
is greater than or equal to the degree of
,
then by long division,
where
is a proper rational fraction; that is, the degree of
is less than the degree of
.
A theorem in advanced algebra states that every proper rational
function can be expressed as a sum
where
are rational functions of the form
or
in which the denominators are factors of
.
The sum is called the partial fraction
decomposition of
.
The first step is finding the form of the partial fraction
decomposition of
is to factor
completely into linear and irreducible quadratic factors, and then
collect all repeated factors so that
is
expressed as a product of distinct
factors of the form
and
.
From these factors we can determine the form of the partial fraction decomposition using the following two rules:
Linear Factor Rule: For each factor of the form , the partial fraction decomposition contains the following sum of m partial fractions:
where A1, A2, . . ., Am are constants to be determined.
Quadratic
Factor Rule: For each factor of the
form
,
the partial fraction decomposition contains the following sum of m
partial fractions:
where A1, A2, . . ., Am, B1, B2, …, Bm are constants to be determined.
I. Integrating Proper Rational Functions
Example 6: Find
.
using
the Linear Factor Rule,
we get
after
multiplying
by
.
If we let
,
then
;
if we let
,
then
.
Thus,
=
.
Example
7: Find
.
by
the Linear Factor Rule,
we get
after multiplying
by
.
If we let x
= 2, then
;
if we let x =
3, then
.
Thus,
.