Integration byPartial Fractions

Claire Gui

Calculus

Elite Prep

When & Whydo we use partial fractions?

도형 can only be done if the degree of the nu-

merator is strictly less than the degree of denominator

for each factor in the denominator we can determine which method we should use for partial fraction decomposition

dx

xxxxx2

24

)3)(2(8635

U substitution //Failintegration by parts //Failtrig substitution //Fail PARTIAL FRACTION !!!

01 Distinct Roots1.1 common type1.2 example & solution

02 Repeated Distinct Roots

2.1 common type2.2 example & solution

03 Non-distinct Roots3.1 common type3.2 example & solution

04 Non-distinct repeated Roots4.1 common type

4.2 example & solution

Four different types partial fractions

도형

dxxx

x)2)(2(

43

break the denominator

Distinct Roots

22)2)(2(43

xB

xA

xxx

)2()2(43 xBxAx

252 Bx

212 Ax

dxxx )2(25

)2(21

CxxCxx 5)2)(2()2ln(25)2ln(

21

cross mutiply

substitute the value of A & B into the original problem

find the value of A & B

find the integra-tion and combine

도형

dxxxxx24

3 42

Repeated Distinct Roots

11)1)(1(42

22

3

xD

xC

xB

xA

xxxxx

)1()1()1)(1()1(42 223 xDxxCxxxBxAxxx

4-0 Bx

3331 DxCxx 222220 DxCxBxAxx

2-2 AAxx

1DC6 DC

25-C

27

D

break the denominator(x² means x has two same roots)

compare the coefficients for dif -ferent degrees of x with A/B/C/D

도형

dx

xxxx )1(27

)1(2542

2

Repeated Distinct Roots

Cxxx

x )1ln(27)1ln(

254ln2

xC

xxx 4)1()1(ln 25

7

find the integra-tion and combine

도형

dxxxxx

)5)(4(542

22

3

Non-distinct Roots

54)5)(4(542

2222

3

xDCx

xBAx

xxxx

)4)(()5)((542 223 xDCxxBAxxx

3332 CxAxx 2220 DxBxx CxAxx 454

DB 455

2CA

445 CADB

545 DB 95

D95

B

32

C34

A

도형Non-distinct Roots

dxx

x

x

x

595

32

459

34

22

)4ln(321

32

24

)4(34

2

2

2

xdUU

xdxdUxU

dxxx

2tan

109

2tan21

59

41

59

11

2

xx

dxx

)5ln(311

31

25

)5(32

2

2

2

xU

xdxdUxU

dxxx

dxx 51

95

2

we have to break it into two parts

U substitu-tion

U substitu-tion

tan-gent inverse law

Cxxxxx

2tan

109

55)5()4ln( 1

52

222

find the distinct roots

521

55512

A

xB

xA

x

521

B

)5ln(521)5ln(

521

xx

도형

dx

xxx22

2

)3(365

Non-repeated Distinct Roots

22222 )3(3)3(65

xDCx

xBAx

xx

365)3)(( 22 xxDCxxBAx

find the integra-tion and combine

3363

50

DBCAB

A

126

DC

도형Non-repeated Distinct Roots

find the integra-tion and combine

dxxx

x

222 )3(

1263

5

3tan35 1 x dx

xxx

2222 )3(12

)3(6

322

2

)3(1613

23

xdU

U

xdUxU

2sin3322

)2cos(134

cos34sec3sec312

sec3

tan3

24

2

2

ddx

x

x

3 3tan 1 x

32 x

36

3

3

332 222

x

x

xx

x

Cxxxx

36

3tan32

3tan35

211

trig substitution

Thank you!