Integration By Parts

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

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

vduuvudv

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.

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

e.g. (i) xdxx cos

e.g. (i) xdxx cos xu

e.g. (i) xdxx cos xu

dxdu

e.g. (i) xdxx cos xu

dxdu xdxdv cos

e.g. (i) xdxx cos xu

dxdu xdxdv cos

xv sin

e.g. (i) xdxx cos xu

xdxxx sinsin dxdu xdxdv cos

xv sin

e.g. (i) xdxx cos xu

xdxxx sinsin dxdu xdxdv cos

xv sin

cxxx cossin

e.g. (i) xdxx cos xu

xdxxx sinsin

xdxii log

dxdu xdxdv cos

xv sin

cxxx cossin

e.g. (i) xdxx cos xu

xdxxx sinsin

xdxii log xu log

dxdu xdxdv cos

xv sin

cxxx cossin

e.g. (i) xdxx cos xu

xdxxx sinsin

xdxii log xu log

dxdu xdxdv cos

xv sin

cxxx cossin

xdxdu

e.g. (i) xdxx cos xu

xdxxx sinsin

xdxii log xu log

dxdu xdxdv cos

xv sin

cxxx cossin

xdxdu dxdv

e.g. (i) xdxx cos xu

xdxxx sinsin

xdxii log xu log

dxdu xdxdv cos

xv sin

cxxx cossin

xdxdu dxdv

xv

e.g. (i) xdxx cos xu

xdxxx sinsin

xdxii log

dxxx log

xu log

dxdu xdxdv cos

xv sin

cxxx cossin

xdxdu dxdv

xv

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

1

0

7 dxxeiii x

1

0

7 dxxeiii x xu

1

0

7 dxxeiii x xu

dxdu

1

0

7 dxxeiii x xu

dxdu dxedv x7

1

0

7 dxxeiii x xu

dxdu dxedv x7

xev 7

71

1

0

7 dxxeiii x xu

dxdu dxedv x7

xev 7

71

1

0

71

0

7

71

71 dxexe xx

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

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

xdxeiv x cos

xdxeiv x cos xeu

xdxeiv x cos xeu

dxedu x

xdxeiv x cos xeu

dxedu x xdxdv cos

xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxexe xx sinsin

xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxexe xx sinsinxeu

xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxexe xx sinsinxeu

dxedu x

xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxexe xx sinsinxeu

dxedu x xdxdv sin

xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxexe xx sinsinxeu

dxedu x xdxdv sin

xv cos

xdxeiv x cos xeu

dxedu x xdxdv cos

xv sin

xdxexe xx sinsin

xdxexexe xxx coscossinxeu

dxedu x xdxdv sin

xv cos

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

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

Euler’s Formula

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

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

cos sinie i

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

cosxe xdx

cosxe xdx

cos sinxe x i x dx

cosxe xdx

cos sinxe x i x dx x ixe e dx

cosxe xdx

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

cosxe xdx

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

111

i xe ci

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

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

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

cosxe xdxOR

cosxe xdx 12

x ix ixe e e dx OR

cosxe xdx 12

x ix ixe e e dx 1 11

2i x i xe e dx

OR

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

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

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

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

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

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

2( ) sinii xdx

2( ) sinii xdx2

2

ix ixe e dxi

2( ) sinii xdx2

2

ix ixe e dxi

2 21 24

ix ixe e dx

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( ) 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

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

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

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