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

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# 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 .

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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

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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.

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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

.