4.7 Partial Fraction Integration
o[d#
Integration Techniques
Power Rule
sum
of functions
onRu1e-wmpositionoffunctims.1ntegrationbgParts-prodwtoffundionsExampkDDeterminewhiohteohniqneisappropriate.CDonotmrKoutt@Hdx2Ofxe2x3Ofxex2dxbypartsbypartsu-substitutionQfx2ex3dx5Ofx2exdx4Ofx2e2xdxu-su.bstitnlimbypartsHwicejbypartsHniet.Dfx2ex2dx8Ofx3ex2dx0fx2t5dxcanEdobypartspoworrwlem0reo1d.L
Division With improper fractions
ong Polynomial
50×2+5×+3
6
4
2×-1
=
×
-1+0
Eg
+01×+06-0
-04×202×-0×21=96×-6×-3
2×-14×26=3×+6*5×+5
2×+3
4×2+4×-1=2×+3
-1
-1×+3
+06×03
.
9
#
Partial Fractions
Integration
The method of partial fractions handles any rational
PKK qk)
Caden
proper
12×3
Pq¥s
,
with
,
.
Long Polynomial Division
Fractions
* improper fractions
1
( quotients)
are polynomials
function
degree is higher in numerator than denominator
-
f 11×2+9×+1=+3
-
.
dx
¥xt6
4×+312×3
-
11×49×+18
#÷
20×2+9×2
012×3
0+20×2015×2
-24×2+18
024×2*18
To
=
S
3×2-5×+6 dx
.
±
3×35
-
5¥
+
6x
+
C
=
x3
-
§x2+6xtC
.
as ''*÷d×
×
¥
2¥
@×3#×÷
.no#ExxEtoxto
-0×201=1
#÷
2-xt1xttzdx-fx2-xt1dx-fhEedu-Xt1du-1dxi.ftuduHn1u1-xz1.Iz.xtn1xt1ItC.HnkHt@HtxIITdxy2x2_2x6Xt3kx3t4x2tOx-5Hx3ETx2tOxQ2x2A6x-x-5o6xtF1Iz-f2xt2xt6y2.3zdx-f2x2-2xt6dx-f23gdxWu-xt3.2xs.zI6x-23lnlxt3ltc.ysddtIE
.Ht±
slim
-2314×+4
more old
&
Adding
. .
]
Examples
@
=
If
+
3
eats
'txtx÷
=
x2÷
set
=
xtxt
x÷±
-
Hz
:
exits
x¥te÷¥s¥2
=
x±Ii¥ t ¥x!
+
=xta¥I÷oext¥÷
xiet.xix.tt
't
it
Subtracting Rational Expressions
=
*5sE¥
-
,÷yxE
=
5*+4×+3
(1+3) ( 1)
X
-
=sxe¥E÷,=3¥¥#
ne@
Partial Fractions
ca#
Integration
proper fractions
g@ We need
to
go
partial fractions decompositions
in reverse from adding
& subtracting
rational expressions
.
Let's consider
rational
the rational function
¥11
:
Deconstruct
the
x2t2*3
.
3×-11×42×-3=3×-11
-
(×t3)(x 1)
factor the
denominator
if
necessary
-
=
A
B
,#+#t
-
reakupfaotorproduotsintosum-AlxItBTx3-geHwmmondenominator@tDCx-11-Ax-AtBxt3Bx.tgrouplmatohvariablesconstant.13zIAtB-At3B_sowesystemofeqnations-8-4Bi.e.eliminatimisnsefd.us
rally
-2=13
Find A :
3=Atf2)
5=A
Ansi
Let's
.
.
,€}+×I±
use
this technique to
much easier to integrate !
integrate pnperfractions
!
Examples] Integrate
.
@K¥I#dx Is ±s+z÷±dx
fxjs×# ,
constant
=
Hxfx# tB×!x¥ 2AxtA+Bx
E⇐aYb EMI
3§
:
:}
-313
¥68543
14=7 A
Find B :
2
D= 64+313
9=12+313
-3=313
It
=
A
.
.
a
-±±d×=,f÷dx
-
Vi 2×+1
Uix -3
du=1dx
2du=2dx
sztxadx
dV÷=2dx{
=2f±du
-
±zf÷dv=
Edie
Zlnlul
-
Idx
lnµ1=21nk3t¥nkxt1t 't
205×1*4 d×=f÷+,¥+
,÷zd×
2)
=Ax(x tB(×
×2 -1×+2
-142×2-1×+2=4×2
-24×+13×-213+42
.
.tk#IEbaMIEgtEnHtEtta
constant
It
Endc
A
:
1=f1)tC
2=C
.
=
ftp.xftx?zdx=Stgn2dx+fE2dx=tnk1.xjt+21n1xHtC=ln1xtt1g+21nkHtC
.