topic techniques of integration. 1. integration by parts 2. integration by trigonometric...
TRANSCRIPT
TOPIC
TECHNIQUES OF INTEGRATION
TECHNIQUES OF INTEGRATION
1. Integration by parts
2. Integration by trigonometric substitution
3. Integration by miscellaneous substitution
4. Integration by partial fraction
TECHNIQUES OF INTEGRATION
4. Integration by partial fraction
xg
xfxH
dxxg
xf
A rational function is a function which can be expressed as the quotient of two polynomial functions. That is, a function H
is a rational function if where both f(x) and g(x)
are polynomials. In general, we shall be concerned in integrating expressions of the form:
DEFINITION
If the degree of f(x) is less than the degree of g(x), their quotient is called proper fraction; otherwise, it is called improper fraction. An improper rational function can be expressed as the sum of a polynomial and a proper rational function.
1x
xx
1x
x22
3
Thus, given a proper rational function:
Every proper rational function can be expressed as the sum of simpler fractions (partial fractions) which may have a denominator which is of linear or quadratic form.
The method of partial fractions is an algebraic procedure of expressing a given rational function as a sum of simpler fractions which is called the partial fraction decomposition of the original rational function. The rational function must be in its proper fraction form to use the partial fraction method.
Four cases shall be considered.Case 1. Distinct linear factor of the denominator
Case 2. Repeated linear factor of the denominator
Case 3. Distinct quadratic factor of the denominator
Case 4. Repeated quadratic factor of the denominator
ii bxa For each linear factor of the denominator, there corresponds a partial fraction having that factor as the denominator and a constant numerator.
Case 1. Distinct linear factor of the denominator
That is, nn2211 bxa
N...
bxa
B
bxa
A
xg
xf
where A, B, …..N are constants to be determined
dx
bxa
N...dx
bxa
Bdx
bxa
Adx
xg
xf
nn2211
Thus,
EXAMPLE: Evaluate each integral.
dxxxx
x
54.1
23
3
2
0 2
2
321
14.2 dx
xxx
xx
Case 2. Repeated linear factor of the denominator nbax If the linear factor appears as the denominator
of the rational function for each repeated linear factor of the denominator, there corresponds a series of partial fractions,
n32 bax
N...
bax
C
bax
B
bax
A
where A, B, C, …, N are constants to be determined.
The degree n of the repeated linear factor gives the number of partial fractions in a series. Thus,
dx
bax
N...dx
bax
Cdx
bax
Bdx
bax
Adx
)x(g
)x(fn32
EXAMPLE: Evaluate each integral.
dyyyy
y23 44
3.1
dx
xxxx
x2345
2
33
1.2
)( 2 cbxax For each non-repeated irreducible quadratic factor of the denominator there corresponds a partial fraction of the form.
Case 3. Non-repeated quadratic factor of the denominator
nnn
nn
cxbxa
MbxaN
cxbxa
DbxaC
cxbxa
BbxaA
xg
xf
2
222
2
22
112
1
111 )2(...
)2()2(
)(
)(
where A, B, …..N are constants to be determined
Thus,
nnn
nn
cxbxa
MbxaN
cxbxa
DbxaC
cxbxa
BbxaAdx
xg
xf
222
22
22
112
1
111 )2(...
)2()2(
)(
)(
EXAMPLE: Evaluate each integral.
dxxxx
xx
)1)(1(
236.2
2
2
dx
xx
x
1.1
2
3
ncbxax )( 2 For each repeated irreducible quadratic factor of the denominator there corresponds a partial fraction of the form.
Case 4. Repeated quadratic factor of the denominator
ncbxax
MbaxN
cbxax
DbaxC
cbxax
BbaxA
xg
xf
)(
)2(...
)(
)2()2(
)(
)(2222
where A, B, …..N are constants to be determined
Thus,
ncbxax
MbaxN
cbxax
DbaxC
cbxax
BbaxA
xg
xf
)(
)2(...
)(
)2()2(
)(
)(2222
EXAMPLE: Evaluate each integral.
dxxx
xxx
32
35
)1)(1(
32.2 222 )1(
.1xx
dx
Evaluate each integral.
dxxxx
xx
32
9236.4
23
2
dxxxx
x
)1)(4)(2(
1812.1
dxxx
x
2)3)(2(
)12(.2
dxxx
x
)4)(12(
1713.3
2
2
dxxxx
xx
)22)(32(
8.5
2
2
dxxx
x
222 )1)(1(
2..6