X2 T04 01 by parts (2011)

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Integration By Parts

Transcript of X2 T04 01 by parts (2011)

Page 1: X2 T04 01 by parts (2011)

Integration By Parts

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Integration By PartsWhen an integral is a product of two functions and neither is the derivative of the other, we integrate by parts.

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Integration By PartsWhen an integral is a product of two functions and neither is the derivative of the other, we integrate by parts.

vduuvudv

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Integration By PartsWhen an integral is a product of two functions and neither is the derivative of the other, we integrate by parts.

vduuvudv

u should be chosen so that differentiation makes it a simpler function.

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Integration By PartsWhen an integral is a product of two functions and neither is the derivative of the other, we integrate by parts.

vduuvudv

u should be chosen so that differentiation makes it a simpler function.

dv should be chosen so that it can be integrated

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e.g. (i) xdxx cos

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e.g. (i) xdxx cos xu

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e.g. (i) xdxx cos xu

dxdu

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e.g. (i) xdxx cos xu

dxdu xdxdv cos

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e.g. (i) xdxx cos xu

dxdu xdxdv cos

xv sin

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e.g. (i) xdxx cos xu

xdxxx sinsin dxdu xdxdv cos

xv sin

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e.g. (i) xdxx cos xu

xdxxx sinsin dxdu xdxdv cos

xv sin

cxxx cossin

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e.g. (i) xdxx cos xu

xdxxx sinsin

xdxii log

dxdu xdxdv cos

xv sin

cxxx cossin

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e.g. (i) xdxx cos xu

xdxxx sinsin

xdxii log xu log

dxdu xdxdv cos

xv sin

cxxx cossin

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e.g. (i) xdxx cos xu

xdxxx sinsin

xdxii log xu log

dxdu xdxdv cos

xv sin

cxxx cossin

xdxdu

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e.g. (i) xdxx cos xu

xdxxx sinsin

xdxii log xu log

dxdu xdxdv cos

xv sin

cxxx cossin

xdxdu dxdv

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e.g. (i) xdxx cos xu

xdxxx sinsin

xdxii log xu log

dxdu xdxdv cos

xv sin

cxxx cossin

xdxdu dxdv

xv

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e.g. (i) xdxx cos xu

xdxxx sinsin

xdxii log

dxxx log

xu log

dxdu xdxdv cos

xv sin

cxxx cossin

xdxdu dxdv

xv

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e.g. (i) xdxx cos xu

xdxxx sinsin

xdxii log

dxxx log

xu log

cxxx log

dxdu xdxdv cos

xv sin

cxxx cossin

xdxdu dxdv

xv

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1

0

7 dxxeiii x

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1

0

7 dxxeiii x xu

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1

0

7 dxxeiii x xu

dxdu

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1

0

7 dxxeiii x xu

dxdu dxedv x7

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1

0

7 dxxeiii x xu

dxdu dxedv x7

xev 7

71

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1

0

7 dxxeiii x xu

dxdu dxedv x7

xev 7

71

1

0

71

0

7

71

71 dxexe xx

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1

0

7 dxxeiii x xu

dxdu dxedv x7

xev 7

71

1

0

71

0

7

71

71 dxexe xx

1

0

77

491

71

xx exe

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1

0

7 dxxeiii x xu

dxdu dxedv x7

xev 7

71

1

0

71

0

7

71

71 dxexe xx

1

0

77

491

71

xx exe

491

498

4910

491

71

7

77

e

ee

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xdxeiv x cos

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xdxeiv x cos xeu

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xdxeiv x cos xeu

dxedu x

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xdxeiv x cos xeu

dxedu x xdxdv cos

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xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

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xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxexe xx sinsin

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xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxexe xx sinsinxeu

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xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxexe xx sinsinxeu

dxedu x

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xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxexe xx sinsinxeu

dxedu x xdxdv sin

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xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxexe xx sinsinxeu

dxedu x xdxdv sin

xv cos

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xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxexe xx sinsin

xdxexexe xxx coscossinxeu

dxedu x xdxdv sin

xv cos

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xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxexe xx sinsin

xdxexexe xxx coscossin

xexexdxe xxx cossincos2

xeu

dxedu x xdxdv sin

xv cos

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xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxexe xx sinsin

xdxexexe xxx coscossin

xexexdxe xxx cossincos2

xeu

dxedu x xdxdv sin

xv cos

cxexexdxe xxx cos21sin

21cos

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Euler’s Formula

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Euler’s FormulaWhen the two functions are a mixture of trig and exponentials, Euler’s Formula can be useful;

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Euler’s FormulaWhen the two functions are a mixture of trig and exponentials, Euler’s Formula can be useful;

cos sinie i

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Euler’s FormulaWhen the two functions are a mixture of trig and exponentials, Euler’s Formula can be useful;

cos sinie i

This leads to;

cos2

i ie e

sin2

i ie ei

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cosxe xdx

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cosxe xdx

cos sinxe x i x dx

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cosxe xdx

cos sinxe x i x dx x ixe e dx

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cosxe xdx

cos sinxe x i x dx x ixe e dx 1 i xe dx

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cosxe xdx

cos sinxe x i x dx x ixe e dx 1 i xe dx

111

i xe ci

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cosxe xdx

cos sinxe x i x dx x ixe e dx 1 i xe dx

111

i xe ci

1 cos sin2

xi e x i x c

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cosxe xdx

cos sinxe x i x dx x ixe e dx 1 i xe dx

111

i xe ci

1 cos sin2

xi e x i x c

cos sin cos sin2

xe x i x i x x c

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cosxe xdx

cos sinxe x i x dx x ixe e dx 1 i xe dx

111

i xe ci

1 cos sin2

xi e x i x c

cos sin cos sin2

xe x i x i x x c

Equating real parts;

cos cos sin2

xx ee xdx x x c

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cosxe xdxOR

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cosxe xdx 12

x ix ixe e e dx OR

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cosxe xdx 12

x ix ixe e e dx 1 11

2i x i xe e dx

OR

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cosxe xdx 12

x ix ixe e e dx 1 11

2i x i xe e dx 1 11 1 1

2 1 1i x i xe e c

i i

OR

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cosxe xdx 12

x ix ixe e e dx 1 11

2i x i xe e dx 1 11 1 1

2 1 1i x i xe e c

i i

1 11 1 12 2 2

i x i xi ie e c

OR

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cosxe xdx 12

x ix ixe e e dx 1 11

2i x i xe e dx 1 11 1 1

2 1 1i x i xe e c

i i

1 11 1 12 2 2

i x i xi ie e c 1 1

2 2 2

ix ixx i e i ee c

OR

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cosxe xdx 12

x ix ixe e e dx 1 11

2i x i xe e dx 1 11 1 1

2 1 1i x i xe e c

i i

1 11 1 12 2 2

i x i xi ie e c 1 1

2 2 2

ix ixx i e i ee c

2 2 2

ix ixx ix ix i e ee e e c

OR

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cosxe xdx 12

x ix ixe e e dx 1 11

2i x i xe e dx 1 11 1 1

2 1 1i x i xe e c

i i

1 11 1 12 2 2

i x i xi ie e c 1 1

2 2 2

ix ixx i e i ee c

2 2 2

ix ixx ix ix i e ee e e c

OR

2 2 2

ix ixx ix ix e ee e e ci

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cosxe xdx 12

x ix ixe e e dx 1 11

2i x i xe e dx 1 11 1 1

2 1 1i x i xe e c

i i

1 11 1 12 2 2

i x i xi ie e c 1 1

2 2 2

ix ixx i e i ee c

2 2 2

ix ixx ix ix i e ee e e c

OR

cos sin2

xe x x c

2 2 2

ix ixx ix ix e ee e e ci

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2( ) sinii xdx

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2( ) sinii xdx2

2

ix ixe e dxi

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2( ) sinii xdx2

2

ix ixe e dxi

2 21 24

ix ixe e dx

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2( ) sinii xdx2

2

ix ixe e dxi

2 21 24

ix ixe e dx 2 21 1 12

4 2 2ix ixe x e c

i i

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2( ) sinii xdx2

2

ix ixe e dxi

2 21 24

ix ixe e dx 2 21 1 12

4 2 2ix ixe x e c

i i

2 21 2

4 2

ix ixe e x ci

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2( ) sinii xdx2

2

ix ixe e dxi

2 21 24

ix ixe e dx 2 21 1 12

4 2 2ix ixe x e c

i i

2 21 2

4 2

ix ixe e x ci

1 sin 2 24

x x c

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2( ) sinii xdx2

2

ix ixe e dxi

2 21 24

ix ixe e dx 2 21 1 12

4 2 2ix ixe x e c

i i

2 21 2

4 2

ix ixe e x ci

1 sin 2 24

x x c

1 1 sin 22 4

x x c

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2( ) sinii xdx2

2

ix ixe e dxi

2 21 24

ix ixe e dx 2 21 1 12

4 2 2ix ixe x e c

i i

2 21 2

4 2

ix ixe e x ci

1 sin 2 24

x x c

1 1 sin 22 4

x x c

Exercise 2B; 1 to 6 ac, 7 to 11, 12 ace etc