Chapter 05 -Solution Manual for Complex Analysis 3th Edition Dennis Zill - Patrick Shanahan

Chapter 5 Integration in the Complex Plane Page |1

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Chapter 5
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Note: Solutions not appearing in this complete solutions manual can be found in the student
study guide. © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
Exercises 5.1
1 3
( 3 − (−1)= ) 1©(27 28
0 2 3 3
2.© Jones
∫ t &
2 Bartlett
dt + ∫ x 2 Learning,
dx + ∫ u 2
du
= LLC
∫ t 2=
dt 3 Jones
1) & Bartlett Learning, LLC
+=
NOT FOR SALE OR DISTRIBUTION 3
−1 0 2 −1 3NOT FOR 3SALE OR DISTRIBUTION

3. If we set u = 2π x, then du = 2π dx or du / (2π ) = dx. By the method of u-substitution we
© Jones & Bartletthave:
Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
1
∫ sin 2π x dx = 2π ∫ sin u du
1
= −
© Jones & Bartlett u
cos Learning, LLC © Jones & Bartlett Learning,
2π
NOT FOR SALE 1 OR DISTRIBUTION NOT FOR SALE OR DISTRIB
= − cos 2π x.
2π

Therefore, by the fundamental theorem of calculus
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION 1 NOT FOR SALE OR DISTRIBUTION
1 1 1
∫1/2 sin 2π x dx =
−
2π
cos 2π x =
1/2
− .
π

© Jones & Bartlett Learning, LLC © Jonesπ& /8 Bartlett Learning, LLC
π /8 π /8 cos 2 x + sin 2 x 1  1 1
4. OR∫ DISTRIBUTION
NOT FOR SALE = ∫
sec 2 xdx =dx NOT
tan FOR
2 x =SALE =[1] OR DISTRIBUTION
0 0 cos 2 2 x 2 0 2 2


1 1 1
ln 3 ln 3
6. ∫ln 2 e −x
dx
© =− e
Jones−x
&=− ( e − ln 3 −Learning,
Bartlett 3− ln 2 ) =
−  LLC
− = © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION
ln 2  3 2 6 NOT FOR SALE OR DISTRIB

7. If
© Jones &=
Bartlett
u x= Learning, LLC
dv e − x /2 dx © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
du = dx v = −2e − x /2

then by the method of integration by parts we have:
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, Chapter 5 Integration in the Complex Plane Page |2

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
∫ xe dx = ∫ u dv
− x /2
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
= uv − ∫ v du
= −2 xe − x /2 + 2 ∫ e − x /2 dx
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
NOT FOR= −2 xe − x /2OR
SALE
− x /2
− 4eDISTRIBUTION
. NOT FOR SALE OR DISTRIB

Therefore, by the fundamental theorem of calculus

4
© Jones & Bartlett− x /2Learning, LLC
4
8 12 © Jones & Bartlett Learning, LLC
∫
− x /2 − x/2
xe dx =−2 xe − 4e =− 2.
NOT FOR SALE OR DISTRIBUTION 2 e e
2
NOT FOR SALE OR DISTRIBUTION


 − e x 1 dx = [ x ln x − x]e = (e − e) − (−1) = 1
e 3
8. ∫1 ln xdx = x ln x
 ∫1 x 1
© Jones & Bartlett Learning, LLC1 © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
4 2x −1 4 2x 4 1 4 2x + 6 4 7
10. ∫2 ( x + 3)2 dx =∫2 ( x + 3)2 dx −∫2 ( x + 3)2 dx =∫2 ( x + 3)2 dx −∫2 ( x + 3)2 dx
4
 & Bartlett
© Jones 7  LLC  7
=  ln( x + 3) 2 + Learning,
= (ln 49 + 1) −  ln 25 + 
© Jones & Bartlett Learning,
 SALE OR
NOT FOR x +DISTRIBUTION
3 2  5 NOT FOR SALE OR DISTRIB
49 2 7 2
ln − = 2 ln −
25 5 5 5
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
11. For CSALE
NOT FOR parametrized
OR DISTRIBUTION ≤ t ≤ πFOR
by x(t ) = 5cos t , y (t ) = 5sin t , 0NOT / 4 we x′(tDISTRIBUTION
haveOR
SALE ) = −5sin t
and y′(t ) = 5cos t. Identifying G ( x, y ) = 2 xy in (6) of Section 5.1 we have:

π /4
© Jones & Bartlett = ∫C G(LLC
Learning, x, y )dx ∫ 0
2(5cos t )(5sin t )(−5sin&t )Bartlett
© Jones dt Learning, LLC
NOT FOR SALE OR DISTRIBUTION π /4 NOT FOR SALE OR DISTRIBUTION
= −250 ∫ sin 2 t cos t dt
0
π /4
= −250 ∫ u 2 du ← u sin =
= t , du cos t dt
0

250 3Learning, LLC
© Jones & Bartlett © Jones & Bartlett Learning,
= − u
NOT FOR SALE 3 OR DISTRIBUTION NOT FOR SALE OR DISTRIB
π /4
250 3
= − sin t
3 0

© Jones & Bartlett Learning,
125LLC
2 © Jones & Bartlett Learning, LLC
=−
NOT FOR SALE OR DISTRIBUTION . NOT FOR SALE OR DISTRIBUTION
6

From (7) of Section 5.1 we have:

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NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION


© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.

, Chapter 5 Integration in the Complex Plane Page |3

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
π /4
∫ G( x, y)dy = ∫ 2(5cos t )(5sin
NOT FOR SALE OR DISTRIBUTION
C
NOTt )(5cos
0
FORt )SALE
dt OR DISTRIBUTION
π /4
= 250 ∫ cos 2 t sin t dt
0
π /4
© Jones &= −250 ∫ Learning,
Bartlett u 2 du ←u =
LLC cos t , du =− sin t dt© Jones & Bartlett Learning,
0
NOT FOR SALE 250OR DISTRIBUTION NOT FOR SALE OR DISTRIB
= − u3
3
π /4
250
= − cos3 t
3 LLC 0
© Jones & Bartlett Learning, © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION
125 NOT FOR SALE OR DISTRIBUTION
=−
6
2 −4 . ( )
From (8) of Section 5.1 we have:
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION π /4 NOT FOR 2SALE OR2 DISTRIBUTION
= ∫ G( x, y)ds ∫ 2(5cos t )(5sin t ) 25sin t + 25cos tdt
C 0
π /4
= 250 ∫ cos t sin t dt
0
π /4
© Jones
= & Bartlett
250 ∫ uLearning,
du ←LLC
= u sin =
t , du cos t dt © Jones & Bartlett Learning,
NOT FOR SALE 0OR DISTRIBUTION NOT FOR SALE OR DISTRIB
= 125u 2
π /4
= 125sin 2 t
0

125 LLC
© Jones & Bartlett Learning, © Jones & Bartlett Learning, LLC
= .
NOT FOR SALE OR DISTRIBUTION
2 NOT FOR SALE OR DISTRIBUTION


12. G ( x, y ) = x 3 + 2 xy 2 + 2 x, x = 2t , y = t 2 , 0 ≤ t ≤ 1
© Jones & Bartlett Learning, LLC 1 © Jones & Bartlett Learning, LLC
NOT FOR SALE OR∫CDISTRIBUTION ∫0 (8t + 4t + 4t )2dt
3 5
6( x , y ) dx= NOT FOR SALE OR DISTRIBUTION
1 1 1 1 
= 8∫ (2t 3 + t 5 +=
t) 8  t 4 + t6 + t 2 
0
2 6 2 
4 1 35
© Jones & Bartlett =Learning,
4 + + =LLC © Jones & Bartlett Learning,
3 2 6
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
1
∫ ∫ (8t
3
6( x, y )dx= + 4t 5 + 4t )2tdt
C 0
1 16 8 8 736
= ∫Learning,
© Jones & Bartlett (16t 4 + 8t 6 +LLC
8t 2 )dt = + + = © Jones & Bartlett Learning, LLC
0 5 7 3 105
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION



© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION


© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.

, Chapter 5 Integration in the Complex Plane Page |4

© Jones & Bartlett Learning,2 LLC © Jones & Bartlett Learning, LLC
x 3
NOT FOR SALE
14. ORGDISTRIBUTION
( x=
, y) 3
2 y 3 x , 1 ≤ x ≤ 8 ⇒ NOT
;= 3/2
≤ y ≤FOR
24 2SALE OR DISTRIBUTION
y 2
8
8x2 2 8 12 22
∫C © Jones
G=( x , y ) dx ∫1 3& 3Bartlett
= dx =∫ x dx
3 1Learning,
x x
3 3LLC 1 © Jones & Bartlett Learning,
NOT FOR x 2
2 SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
4
= (16 2 − 1)
9
4
∫
G
© Jones(& Bartlett
=24 2
y )dy − ∫Learning,
x, = G ( x=
b 8=
LLC G ( x), f=
, y )dy ∫
b1 24 2
( x)dx ∫ © Jones G ( f −& y ) dy (16 2 − 1LLC
1 Bartlett Learning,
( y ),
= )
C 3/2
=
NOT FOR SALE OR DISTRIBUTION a 1=
NOT FOR SALE OR 9DISTRIBUTION
a1 3/2




15. For curve defined by y= x + 3, −1 ≤ x ≤ 2, we have dy = dx, and so

© Jones & Bartlett Learning, LLC 2 © Jones & Bartlett
2 Learning, LLC
∫C (2 x + y)dx + xy dy
NOT FOR SALE OR DISTRIBUTION
= ∫−1 ( 2 x +NOT
( x + 3) ) dx + ∫ x( x + 3)dx
FOR SALE −1 OR DISTRIBUTION
2
= ∫ −1
( x 2 + 6 x + 3)dx
2
x3
= + 3 x 2 + 3 xLLC
© Jones & Bartlett Learning, © Jones & Bartlett Learning,
3
NOT FOR SALE OR DISTRIBUTION −1 NOT FOR SALE OR DISTRIB
= 21.

2
16. ∫ = ∫ (2 x + x 2 + 1)dx + 2( x 2 + 1)2 xdx
(2 x + y )dx + xydy
© Jones & Bartlett Learning,
C −1 LLC © Jones & Bartlett Learning, LLC
NOT=FOR SALE
2 OR DISTRIBUTION 2 NOT FOR SALE OR DISTRIBUTION
∫ ( x + 2 x + 1)dx + (2 x + 2 x )dx= ∫ (2 x + 3x + 2 x + 1)dx
2 4 2 4 2
−1 −1
2
2 5  64   −2  141
x + x3 + x 2 + x =  + 8 + 4 + 2  −  − 1 + 1 − 1=
= 
5 −1  5   5  5
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NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
C1 : x = −1 y : 2  0
18. ∫C Pd x + Qd y = ∫C1 + ∫C2 + ∫C3 C2 : y = 0 x : −1  2
C3 : x = 2 y : 0  5
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
0
NOT FOR SALE 0 OR DISTRIBUTION −1 2 NOT FOR SALE OR DISTRIB
= ∫ ( 2(−1) + y ) dx + (−1) ydy
∫C1 (2 x + y)dx + xydy = y ,=dx 0
2 2 2

2 2 2

∫C
= ∫ (2 x + 0)dx + x(0)dy = ∫ 2 xdx = x 2 = 4 − 1 = 3, dy = 0
© Jones2 & −Bartlett
1 Learning, LLC −1 −1 © Jones & Bartlett Learning, LLC
NOT FOR SALE5
OR DISTRIBUTION 5 NOT FOR SALE OR DISTRIBUTION
∫ =∫ (4 + y)dx + 2 ydy = y =25, dx =0
2
C3 0 0

= 2 + 3 + 25 = 30 ∫
C
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION


© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.