4.7 partial fraction integration - stem math &...
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This was created by Keenan Xavier Lee, 2013. See my website for more information, lee-apcalculus.weebly.com.
4.7 Partial Fraction Integration
This was created by Keenan Xavier Lee, 2013. See my website for more information, lee-apcalculus.weebly.com.
o[d# Integration Techniques
Power Rule sum of functions
iisubslilnlionRu1e-wmpositionoffunctims.1ntegrationbgParts-prodwtoffundionsExampkDDeterminewhiohteohniqneisappropriate.CDonotmrKoutt@Hdx2Ofxe2x3Ofxex2dxbypartsbypartsu-substitutionQfx2ex3dx5Ofx2exdx4Ofx2e2xdxu-su.bstitnlimbypartsHwicejbypartsHniet.Dfx2ex2dx8Ofx3ex2dx0fx2t5dxcanEdobypartspoworrwlem0reo1d.L
ong Polynomial Division With improper fractions
�4�
4×2+4×-1=2×+350×2+5×+3 × -1+0
2×-1 6=
Eg
2×+3-1
2×-14×26=3×+6*5×+5-04×202×-0×21=96×-6×-3-1×+3
+06×03
+01×+06-0.
9
This was created by Keenan Xavier Lee, 2013. See my website for more information, lee-apcalculus.weebly.com.
# Partial Fractions Integration
The method of partial fractions handlesany
rational function,
Pq¥s,
with
PKK qk) are polynomials (quotients) .
Caden properFractions Long Polynomial Division
* improper fractions - degree is higher in numerator than denominator.
�1�
f12×3 -
11×2+9×+1=+3 dx
¥xt64×+312×3 - 11×49×+18
012×3#÷20×2+9×20+20×2015×2
-24×2+18024×2*18
To.
=
S 3×2-5×+6 dx ±
3×35 - 5¥ +6x + C = x3 - §x2+6xtC .
This was created by Keenan Xavier Lee, 2013. See my website for more information, lee-apcalculus.weebly.com.
as ''*÷d××
.no#ExxEtoxto@×3#×÷2¥-0×201=1¥
#÷
tSx2-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.ysddtIEfxdx.a
.Ht±slim-2314×+4
This was created by Keenan Xavier Lee, 2013. See my website for more information, lee-apcalculus.weebly.com.
more old. .. Adding & Subtracting Rational Expressions
Examples]@ If + xtxt = eats'txtx÷ = x¥te÷¥s¥2
=
x2÷ itset x÷± : exits + x±Ii¥⇒t ¥x!=
xiet.xix.tt =xta¥I÷oext¥÷�3� 't - Hz = *5sE¥ - ,÷yxE⇒ = 5*+4×+3
(1+3) (X - 1)
=sxe¥E÷,=3¥¥#
ne@ Partial Fractions Integration
ca# proper fractions partial fractions decompositions
g@ We need to go in reverse from adding & subtracting rational expressions.
This was created by Keenan Xavier Lee, 2013. See my website for more information, lee-apcalculus.weebly.com.
Let's consider the rational function : ¥11 Deconstruct therational
.
x2t2*3
3×-11×42×-3=3×-11
- factor the denominator if necessary(×t3)(x - 1)
A B= ,#+#t
-
breakupfaotorproduotsintosum-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 ,€}+×I± much easier to integrate !
Let's use this technique to integrate pnperfractions !
This was created by Keenan Xavier Lee, 2013. See my website for more information, lee-apcalculus.weebly.com.
Examples] Integrate.
@K¥I#dx Is ±s+z÷±dx
⇒fxjs×#,= Hxfx#⇒tB×!x¥ 2AxtA+Bx -313
constant :} E⇐aYb⇒3§:EMI ¥68543
14=7 AFind B: D= 64+313 2 = A
.
9=12+313-3=313
.
Ita -±±d×=,f÷dx - sztxadx
Uix -3 Vi 2×+1du=1dx
dV÷=2dx2du=2dx Edie Idx
=2f±du - ±zf÷dv= Zlnlul - { lnµ1=21nk3t¥nkxt1t 't
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205×1*4 d×=f÷+,¥+ ,÷zd×
⇒ ×2 -1×+2 =Ax(x⇒tB(×. 2)-142×2-1×+2=4×2-24×+13×-213+42
constant.tk#IEbaMIEgtEnHtEttaIt A
Endc :
1=f1)tC2=C
.
=
ftp.xftx?zdx=Stgn2dx+fE2dx=tnk1.xjt+21n1xHtC=ln1xtt1g+21nkHtC
.