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```A 303

113

917

nsuer

book

opy 1

303

ANSWER BOOK

17 TO ACCOMPANY

suer MARCH AND WOLFF'S CALCULUS

°° k BY VyJ^

py 1 HERMAN W^ilARCH

and y

HENRY C>"WOLFF

Copyright, 1917, by the McGraw-Hill Book Company, Inc.

Page 11, §12

1. (a) y =2(x - 2)2; y = 2(x + 3) 2 ; y = 2x 2 + 5; y - 2x 2 - 1;

y = 2(x + 2) 2 - 1.

(b) y = - 3(s - 2) 2 ; y = - 3(x + 3) 2 ; y = - 3a; 2 + 5;

y = -3x 2 - l;y = -3(x + 2) 2 - 1.

(c) ?/ = log (x - 2); y = log (x + 3); y = log a; + 5;

2/ = log x - 1; y = log (x + 2) - 1.

(d) y = e~ x+2 ; i/ = e" 35 " 3 ; y = e~* + 5; y = e~ x - 1;

2/ = e~ x ~ 2 - 1.

2. (a) y = 2x 2 + §x; 2/ = 2x 2 - £x; y = 2x 2 + x; y = 2x 2 - x.

(b) y = - 3x 2 + \x\ y = - 3x 2 - \x\ y = - 3x 2 + 3;

?/ = - 3x 2 _ x>

(c) ?/ = log x + \x\ y = log x - \x) y = log x + x;

2/ = log £ — x.

(d) y = e~ x + \x\ y = e~ x - Jo:; 1/ = e~ x + x; y = e~ x - x.

3. (0) y - - 2x 2 ; y = 2x 2 ; x = 2?/ 2 ; x = - 2y 2 .

(b) y = 3x 2 ; y = - 3x 2 ; x = - 3?/ 2 ; x = 3?/ 2 .

(c) ?/ = log - ; y = log ( - x) ; x = log 2/; x = - log ( - y).

(d) y = — e~ x ; 1/ = e z ; x = e~ y ) x = — e y .

4. (a) p = a sin ( — ~) ; p = a sin f — ^j ; p = — a cos 0;

p = — a sin 0; p = a cos 0.

(6) p = a cos ( — o) ; p = a cos (0 — A ; p = a sin 0;

p = — a cos 0; p = — a sin 0.
6. (a) 2x7/ = a 2 ; (6) 2x?/ = - a 2 ; (c) y 2 - x 2 = a 2 ;

(d) y 2 - x 2 = a 2 ; (e) x 2 + y 2 = a.

Page 26, §20

j 1 . - *

" 2\/x - 2

/ft

f

RESULTS OF EXERCISES Yx^

Page 39, §30 ^\\*V 9^

1. 0.27 of 1 square inch per second. ^ O* 5 ^

2. 1.56 square inches per minute. O^*

3. 0.0051 of 1 inch per minute; 0.8 of 1 square inch per minute.

Page 42, §32

12. 14a; (5 - 2x 2 )~%

15. -±±x(x 2 + 1)'

17. - 8x(x 2 + 4)" 3 .

18. 18o; 2 (5 - a; 3 )" 3 .
Page 46, §36

3. * 4.

X

y

b 2 x

5. — —k -'

a 2 y

'• - (?)'•

Page 51, §37

v&

39. - i(4 - 3a; 2 ) 2 + C. 41. - A(4 - 3a;) 2 + C.

42. f(o; 3 + 3a; - 7)* + C. 45. J (6 + 4z - 10a; 2 )* + C.

48. - J(2 - a; 2 ) 4 + C. 49. 3y = a; 3 + 1.

50. 32/ = 2a;* + 4(3 - a/2). 51. xy = 1.

Page 55, §38

1. 1610 feet. 2. 1710 feet; 332 feet per second.

3. 257.6 feet down; 966 feet below the starting point; 1094.8 feet.

4. (1°) v = Z 2 ; s = \t\

(2°) v = t 2 - 2; s = %t* - 2t + 3.

Page 58, §40

^ 2x 2 + x - 2 12 + 4a; - 5a; 2

9. 7=-- 10. -5— •

V a; 2 - 2 3(3.2 _ 4) s

8 - 3a; - 4a; 2 1 - 2a; 2

11. 7 — 12.

\/4 - a; 2 Vl

a;

13 dy = 3(l + ^) _ 14 ^ = y {2x " %) .

'da; 7 — 3a; 3 ?/ 2 * " da; a;(6?/ — x) '

di _ 2yVx + yVy 16 ^ _ y(2x - 1)

^ 2*V|j + * V* " dx x{l ~ x)

Page 59, §41
-2 (z-3)(s-l) . 1 + 2a; - x 2

{x-l) 2 ' (x-2) 2 . "" (1-a;) 2

JUL -2l9l7©aA467695.

3- x

RESULTS OF EXERCISES

3x 2 + i

7.

2(1 +x)Wx- 1
x + \/x 2 - 1

-sA

- 12s

(x 2 +l) 2 '

5.

8.

2x*

4x — 4

(a; -2)2
Page 60, §42

8. - 7x(x 2 + 1)~-
Page 62, §63

6.

9.

2 + x

2(1- x) f

2(x 3 - 3x 2 -

-4)

(x - 2)2

9. x(l - x 2 )

Ex.

Increasing when

Decreasing
when

Maximum
value

Minimum
value

1.

x >
x < and x >

x <

2.

3.

3 <

s >

4.

For all values of x

5.

x < and x > f

< a? < |

- 6

6.

x > J

x < §

2 5

7.

x < 1 and a: > 2

1 < x < 2

- 5

- 6

8.

s < 1 and s > 1

- 1 <3 < 1

9

5

9.

x < — 1 and a > ^

- 1 < X < %

— #1

10.

3 <

x >

11.

13.
14.

17.

21.

24.

40.
47.

4 inches. 12. 2.4 feet by 5 feet.

When t — 4; Approaching at the rate of 5 miles per hour; Sepa-
rating at the rate of 7.905 miles per hour.
6 inches by 6 V3 inches.

Page 63, §63

(x - 1) (3x - 1). 18. s(2 - 3x). 19. (1 - x 2 ) 2 (1 - 7x 2 ).

20. 2(1 - x 3 ) (x - 2) (1 + 6x 2 - 4x 3 ).

l + 2x

(x 2 + l) 2
a 2

(a 2 - x 2 )^'

22. - ,. , „. x . x 23. i(x - 1) 3 .

25.

(1 + x 2 ) 2
1- 2x 2
Vl - x 2 '

39. Kl -sT'+C.

- \{2x - x 2 ) + C. 43. - 2vT- x + C.

46.

9x
16y

48. f V3 feet per second.

RESULTS OF EXERCISES

49. 0; 0.201; 2.27; oo . 50. 18.08.

51. x = 14.14; angle = 45°. 52. - gt cot 20° = - 88.4*.

53. 0; - 494.3; - 695. 54. 18.32.

55. 12.53; 13.96; A of one hour. 56. 9; 27.58; when t = 1.484.

57. 38.43. 58. 500. 59. 249.5. 60. - 840.

61. 2tt; 54; 2.16.

1. (a)

2. (a)

Su

VV + 7

y

2(y* - 3)*

32/
7y — 4z 3
4?/ 3 - 7x

62. 11.33; 15.1

2.86;

153.7.

Page 67, §45

b)

2(6u 2 '+ 5) (x -

-1).

(c) -

2xu
(u* - 5)*

»-!

1/2+2)3

4y

(d) 1.

(b) -

3?/ 3 + lSx 2 y + Sx
9xy 2 + 6z 3

Page 68, §46

2.

- /2-n.

n l
Page 69, §47

3. -

- iu~K

(c) ■
3. (a)

1. i.

1. 2V2; 2V2; 2; 2.

2. y = x + 1, equation of tangent line;

y = — a; + 3, equation of normal line.

3. 9x+ 20y = 75. 4. y* = 8x + 1.
5. V5. 6. iVl7. 7. 2/ 2

IV17.
Page 74, §49

2x +7.

Ex.

Maximum
point

Minimum
point

Point of inflection

1.

2.
3.

(0,0)

(-2,4)

(2, - 4)
(0,0)

(1, - 2)
(-1,2)
( - t, 5)

4.

(1, - 2)

(0, 1) and (f, - *f )

(0,0)

(_ i _ 13 1) an d (| 18 J)

5.

6.

(-1, -35f)

\2l 87

7.

(0, 1) and (i If)

RESULTS OF EXERCISES

Page 77, §50
1. fm; ¥; 10J. 2. }; |a 2 ; J; T V 3. 2.797.

Page 81, §51

1. 20 foot-pounds. 2. 1136 foot-pounds. 3. 119 ergs.

4. 8920 foot-pounds. 5. 20.83 foot-pounds.

Page 87, §54
1. 45°.

Page 99, §62

5. M* 2 - 2)~hx. 6. 7^-~ dx -

2y/2{x - 1)3

7. (x + 1) O - l) 2 (ox + l)dz. 8. S(x 2 + x - 2) 2 (2x + l)dx.

9. - i(x - l)~-dx. 10. - Sx(x 2 - l)~ f dr.

19. y = Cx. 20. ?/ = CSA 2 - 1.

21. i/ = C(x 3 - 3z 2 + 12x - 2)3.

Page 102, §63

z 2cte

3. Vl + 4a: 2 dx. 4. -^ 1 + JL

efo.
4x

T.^/l+^dx. 8. \jl+^dx.

Page 105, §65
1. 14. 2. f. 3. ia 3 (2\/2 - 1).

Page 109, §67

2. 121,100 foot-pounds. 3. 21,880 foot-pounds.

4. 160 inch-pounds. 5. 1012.5 inch-pounds.
6. 19,475m + 3950 foot-pounds.

Page 110, §68

1. 27tt. 2. *irr\ 3. 4Stt; 64tt. 4. 64tt.

5. ftWo». 6. Va 3 . 7. }tt.

RESULTS OF EXERCISES
Page 112, §69

1.

1.736.

2.

6a. 3. a

J Va 2 - x 2

4.

16.67.

Page 113, §70

1.

16V6tt.

2. <

txa 2 . 3. ^tt.
Page 116, §72

4. -\ 2 -ira 2 .

1.

6000; 18,500 pounds.

2. 5208 pounds.

Page 121, §74

1.

JV3. 2.

1.724.

3. 6. 4. 30.

5. i

6.

17.9. 8.

5k.

9. 2g. 10. 100.

11. 6|; 7§

L2.

-r , where r is the radius.

Page 130, §76

^~ ~ . ~ , n ^ . „ x -, - ~ 2 cos 2 2x-\- 4 sin 2 2x ,

12. 2 sin 2z(sec 2 2x + l)dx. 13. f^ dx.

K J cos 3 2x

14. (2s + 3) cos (x 2 + dx - 2)dx.

15. 2(3 - x) sin (3 - x) 2 dx.

cos 2s

16. / . ~~~ dx. 17. (cos 2x — 2x sin 2x — 2 sec 2 2x)dx.
V sin 2x

20. cot* (2a? - 1) [cos 2 (2s - 1) - 3 sin 2 (2x - l)]dx.

31. 2. 32. Tra 2 . 33. 8a.

34. 37ra 2 . 35. Tra 2 . 36. irab.

37. - tan 0; 6a. 38. Jtt 2 .

39. 4.37 feet per second. 40. 5.364 feet per minute.

41. 2 ^Z'K 42. - 4?-.

Sy-2 cos y

44. 3a* cos 30 + ,,' sin . _ c _

2p cos 1 6 L J

46. ^[sin 7x + 7 sin z] + C.

47. Jy[7 cos 3x - 3 cos 7x] + C.

48. T io[H sin 5z - 5 sin lis] + C.

49. ^[11 cos 3z - 3 cos lis] + C.

50. 5 i 6 [2 sin Ux + 7 sin 4s] + C.

51. 7^^-^ £T [(co — a) cos (a> + a)£ + (a> + a) cos (w — a)t] + C.

RESULTS OF EXERCISES
1

-^ [(co — a) sin (co + a)t + (co + a) sin (co — a)* 1 ] + C.
53.

"• 2(co 2 - a 2 )
2

Page 134, §77

7.

- 1
2a: 2 - 2x + 1

2(*-l)

' Vi - (i - x) 4

9.

2x

X

(x 2 - 3) V* 4 -

6x 2

10. / 4- sin * x.

+ 8 Vl - z 2

11.

itan-i| +C.

15. § tan" 1 x 2 + C.

16.

fsin" 1 2x +C.

2r
17. | sin" 1 -3- + C.

19.

| sin" 1 a: 2 + C.

AFT ^

37 '6-

38.

2wa 2 .

39. 0.848.
Page 139, §79

2. 250tt feet per second.

3. — IOOtt radians per minute per minute; 6 J revolutions.

4. 200tt revolutions.

v 2

5. rco 2 radians per second per second, or — feet per second per

second, where v is the tangential velocity; r is measured as feet and
time as seconds. The acceleration is directed toward the center of
the wheel.

6. rco feet per second, if r is measured as feet and co as radians per
second. The point is moving in a direction perpendicular to the
line joining it and the point of contact of the wheel with the hori-
zontal track.

7. (x - b) 2 + (y - c) 2 = a 2 .

Page 144, §81

1. (a) |t; 2 + *y

= 0. (b) Itt; d2 d y t2 + 9y = 0.

2. (a) y = A: sin (3*

- a). (b) y = k sin (\/3£ — a).

Page 148, §82

7 /,Z^n«o ~ c\ \

iA 2x(7+:r 2 )

1 - X 4

8 RESULTS OF EXERCISES

14. e~ x (cos x — sin x). 18. £ 4 5 x (5 + % log 5).

34. log (e x + e~ x ) + C. 35. 10 log 2 = 6.93.

37.

y = Ce*.

38. 0.4343.

39.

— log (esc x + cot a:

) +'C.

46. 0.732.

49.

(0.4343) (10)* + C.

51. log| = 0.451.

55.

3(a 2 +z 3 ) ^ U *

63. log 2 = 0.693.

Page

154, §83

a;- 119

(3 + 3) (133- Sx- 5x 2 )

1.

35(z + 1) (a; - 5)*

(x - 4)\x - 5) 5

Q

17x + 29

A 2 + x - 5x*
4.

6(z + l)'(2a; + 5) 4 2 ^ 1 ~ x

5. x n ~ l n x {n + £ log n). 6. x sin x 1- cos x log a; •

7. 10 3 '- 2 [3(7£ + 3) log 10 +7].

8. \x S//x ~ h {2 + logz). 9. 7. 10. -•

11.

12. -• 13. -• 14. k. 15. h log 1(

XXX

16.

\ du \ dv 1 du 1 dv 1 dw
u dx v dx u dx v dx w dx

19.

y = Cx\ 20. y = Cx\ 21. Ce fF{x)dX . 22. 2/ = Ce*

Page 158, §84

1.

. 1, &2

g<r*«. 3. ^log^jno.

4.

-oe-K 5. , "° •

Page 159, §85

1. Joe" - 02 '.

2. 8.51, assuming that the medium surrounding the body is at zero
degrees temperature.

Page 165, §89

1. f. 2. 2i 3. ft. 4. 16.1; 1.

1. — tan -1 m;

RESULTS OF EXERCISES

Page 168, §90

Wjsin ft — m cos ft)

Page 169, §92

1. Maximum value = 2; Minimum value = — 9.

2. A square of 200 units.

Page 175, §96

1. 7.92 inches. 2. 2\ miles from A. 3. 45°.

4. 19.73 feet. 5. 11.23 feet, 6. 24 feet.

9. -• 10. 66° 3.6'.

7T

11. 6\/3 inches by 6 inches. 12. 4\/6 inches by 4\/3 inches.
14. Part 8, |.

Page 181, §97

1. - 6. 2. B. 3. -• 4. * = I

a 2

5. * = 7T - 2 «

6. * = ^ + e -
Page 182, §98

a'

1. 2;ra.

2. 8a. 3. |

\wa.

^^^-l,;

a l J

Page 183, §99

1. Tra 2 . ' 2. 67ra 2 .

3. 67ra 2 . 4. 2 2 5 7T.

5. * IT.

6. ,Wa 2 . 7. 5.

ft 4 3.2 Q 10r >

8. 37r 3 a 2 . 9. — i—

IT 4

Page 187, §100

10. fc»

28. p - \t* + t + 2 log (« + !) + C.

42. yg5 log V7 + Vto + c
70 - N : - v/5* '
69. i l..« Bee. (2*« - 5) + C. 71. J tan" 1 (i sin x) + C.

73. a*X - EaV + J<W - \x* + C.

10 RESULTS OF EXERCISES

84. | 87. log (a; 2 + 9) + tan" 1 | + C.

88. ^ 2 + 3a; - f log (x 2 + 9) - V tan" 1 ! + C.
102. x - 2 log (2a; + 7) + C.
117. § log (rf - 9) + f log ^| + C.

Page 191, §101

1. sin"* f^) + C. 2. | tan-i (~p) + C.

3.ilo g J^f + C.

#v /o __

4. -~ log [2a; + 1 + V±x 2 + 4x - 6] + C.

7. ^ sin- *£» + C. 8. ^ log ^^ + C.

9.

^ log [2/ + 1+ V^ 2 + U + 2] + C.

2

10. log (a; 2 + 6a; + 25) - V tan" 1 ir~^\ + C

11. 2V2z 2 + 2x - 3 + ^p log [(2a; + 1) +V4a; 2 + 4a; - 6] + C.

12. 2 log (a; 2 + 2x + 5} + I- tan" 1 (^^) + C.

13. - 2\/5 - 4a; - a; 2 + 5 sin" 1 (^p) + C.

14. - fVSTSn^ - ^ sin" 1 (^) + C.

. \/2 . _, / 4 - 3a; \ , „

15. - -2- sin '(-^J + C.

1C V3, 3 (a; + 1) + V^a; 2 + 18a; + 9 , n

16. -T^log- -— +C.

17. sin" 1 (^p) + C.

18. J log [2y + 3 + VV + 12y - 7] + C.
19.^-log^ + C. 20.^ sin" 1 (^) + C.

RESULTS OF EXERCISES 11

21. - V2 + 3.r - z 2 + 4 sin" 1 ( 2x ~ 3 ^ + C.

OQ x , 2 + 3z + V4 + 12z - 7x 2 „
23. — i log h C.

Page 192, §102

1. xfg (8a: 2 - 18s + 81) (2x + 3)* + C.

2. |[Vx + 1 - 3] (x + 1)* + 4 tan" 1 (a + 1)* + C.

3. 2VF+1 + -^log ^^ + I 3 -^ + C.

V3 Vz + 3 + V3

4. Wx - 2 + 3\/2 tan" 1 \^-y^ + C.

5. 6 VT+1 - 4 log V + * — - + C.

V a; + 1 + 1

6. x + 1 + 4Vx + 1 + 4 log Wx + 1 - l) + C.

7. 2 (6x* - Sx* + 2x&) - / f log (3x& + 1) + C.

8. Mb* - 16) (x + 4)* +C.

9. Hog ^L±^ 2 -f C.

V a; + 4 + 2

10. 3 V2T+~ 3 - 4 log y^l^ + C.

V 3 V2x + 3 + V3
11. ^[16x* + 15x*]a; + C.

12. §\/2a7^3 + \,°V3 tan" 1 \^jp^ + <?.

13. tVIGO/ 6 - 84/ 4 + 105J 3 + 140* 2 - I20t - 420]* + 3 log (t 2 + 1)

+ 6 tan" 1 t + C, where t = (x + 2)*

14. 2VaT^3 - 2\/7 tan" 1 yj x ~-- + C.

15. ¥fr A - 3**] + 5 log (1 + x J ) + 10 tan" 1 x'° + C.

16. W2T+3 + ^ log V3V2T+3 - Vg c

9 VsV2x + 3 + Vl3

12 RESULTS OF EXERCISES

Page 194, §103, (a)

1. — cos x + f cos 3 x — i cos 5 x + C.

2. J sin 3 x — \ sin 5 x + C. 3. i sin 4 a; — J sin 6 x + C. .

_3_ 7.

4. f cos 3 # — cos x + C. 5. f (sin a;) 2 — f (sin x) 2 + C.

6. + cos 7 a; - i cos 5 x + C, 7. 2 (sin a;) 2 - |(sin a;)* + C.

8. - f(cos 0) 3 + i(cos 0)* - A (cos 0)^ + C.

9. J sin 3 a. — \ sin 5 a + C.

10. - | cos 3 (2x + 3) + j\ cos 5 (2a: + 3) + C.

Page 195, §103, (b)

1. i(2x - sin 2x) + C.

2. A (24a; + 8 sin 4a: + sin 8a;) + C.

3. T fa(12a; - 3 sin 4a: - 4 sin 3 2a;) + C.

4. Jg (12a; - 8 sin 2a; + sin 4a;) + C.

5. A (6a; - sin 6a;) + C.

6. T k(60a; + 8 sin 10a: + sin 20a;) + C.

Page 196, §103, (c) and (e)

1. i sec 4 x — sec 2 x -\- log sec x + C.

2. — i cot 3 x — cot x + C.

3. | tan 3 x + | tan 5 a; + I tan 7 a; + C.

4. | sec 7 a; — t sec 5 a; + C. 5. — J cot 3 a; + cot a; + x + C.

6. — cot x — f cot 3 a; — i cot 5 a; + C.

7. i tan 5 a: + C.

8. tan a; + f tan 3 a; + \ tan 5 a; + C

9. ^ sec 7 x — f sec 5 a; + J sec 3 a; + C

10. \ tan 7 a + J tan 9 a; + C.

11. | (tan a;) 3 + T 3 T(tan a?)"^" + C:

12. | tan 3 a: + C.

13. | (tan x)* + f(tan »)* + A (tan x)V + C. ■

14. |(sec 2 (9 - esc 2 (9) - log sin + log cos + 2(9 + C.

15. tan - + C.

16. i cot 3 19(3 tan 4 (9-6 tan 2 - 1) + C.

Page 199, §104

1. log (x + Va; 2 - a 2 ) + C.

2. 1 ^ 2 da; = lo S (^ + A

/a 2 + x\

3. AOa; 2 - 2) (1 + x*)* + C.

RESULTS OF EXERCISES 13

4. 1 log 1= + C.

a a + V a 2 + x 2

5. X log ?- + C.

a a + V a 2 - a: 2

6. -4= log 7 _ + C.

V7 V7 + V7 - 3z

jVx 2 + 9

8. * + 1 + C. 9. , *~ — . + C.

VI ~ z 2 V 4x - 3 - a; 2

10. fa 2 [j sin 3 20 - i sin 4(9 + id] + C, where (9 = sin" 1 l^j 3 -

11. ^[sin- 1 - - -Va 2 - x 2 ] + C.

2 a a 2

12. ^[9sin-i^ + V§*V9 - 5**] + C

1U o

13. ^[sec-i | + lV^^9] + C.

14. V[3 sin-i| + ^(45 - 2x 2 )v / 9~^ 2 ] + C.

16. A -7 -- +C.

V 16 - x 2

17. ^ — ^±JL= +C.

Vi ! + 6x + 25

18. j[27 sin-* ^~ - (x + 9) V6x - x 2 ] + C.

19. - J ;. x+ = + c.

Vz 2 + 4x - 5

20. sin" 1 -^J + C.

Vn

Page 201, §105

1 2 *3,r 9 7F . <* 9_ 4. 7F .

1. T6-7T. *• 2 6 - **• 4 * O

a/2
5. 2 4 V. 6. >. 7. ^-f. 8. 0.0855.

9 - w- 10 - i£ "■ **■ 12 - l {b2 - y2) -

14 RESULTS OF EXERCISES

Page 202, §106
1. -q(3 log x — 1) -f C. 2. x sin x + cos x + C.

3. re sin" 1 a; + Vl - x 2 + C. 4. 3Ve 4 *(8a; 2 - 4s + 1) + C.

5. a; tan -1 a; — log Vl + a; 2 + C.

6. sin a; — x cos a; + C.

7. ~^ 2 [(r* + l)k>ga;-l]+C.

8. ij [16a; 3 tan" 1 2a; - 4a; 2 + log (4a; 2 + 1)] + ft

9. % sin a; (12 — sin 2 x) — \x cos a;(3 + cos 2 x) + C.

10. a; (log a: - 1) + C.

11. J (cos 2a; + 2a; sin 2a; - 2a; 2 cos 2x) + C.

12. cos x{l — log cos x) + C.

Page 203, §106

8.602

sin (7* - a) + C, where a = 125° 32'.

2. ttvtt cos (8« - a) + C, where a = 110° 34'.
8.544 '

3. ^j sin (3* -a)+C } where a = 99° 28'.

4. -f^ cos (4* - a) + C, where a = 92° 50'.
4.UU0

e~ x

5. — -=. sin (a; — a) + C, where a = 135°.

V2

6. -r^r cos (5* - a) + C, where a = 121°.

g-0.4*

7. — ; sin (cot — a) + C.

Vco 2 +0.16

e -0 .2*

8. — -. COS (a>£ — a) + C.

V" 2 + 0.04

9. Y^T cos (5* - a) + C, where a = 94° 35'.

10. |^- sin (U -a)+C, where a = 94° 17'.

Page 204, §108

1. — % [esc x cot x + log (esc a; + cot x)\ + C.

2. i sec 3 a; tan x + Msec ai tan a; + log (sec a; + tan a;)] + C.

J. a 4 I sec 5 a

da + C, where a = sec -1 -.
a

RESULTS OF EXERCISES 15

4

a 2 I sec 3 a da.
Jo

5. a 2 I sec 3 a da + C, where a = tan x L .

J

/# — 2
sec 3 a da + C, where a = tan -1 — j=—
V7

7. a 4 I sec a da.

Jo

8. 9 I (sec 3 a — sec a)da.

Jo

Page 208, §109
1. tV 2. 5«ftir. 3. Hf. 4. Att. 5. sV.

6. .¥^. 7. Iff. 8. if. 9. rf 5. 10. A-

11. ~. 12. £. 13. sfer. 14. rf*.

15. T 3 67ra 4 . 16. J-Tra 6 . 17. ^wa\ 18. T f 5 a 3 .

19. Iwa 2 . 20. Jtt. 21. fTra 4 . 22. WaK

Page 211, §110

2+tan ^ / x\

1. i log + C. 2. § tan" 1 (2 tan ^) + C.

o _ + _ x \ 2/

2 - tan ■

tan - — 2
3. J log +C.

2 tan | - 1

1 -^tai —7—

6. - i log (1 - 3sini) + C.

tan * + 2 - V3

7. 2x + * log - - + C.

V3 tan? +2 + a/3

f2 tan % + 1~|

- — Jr- - t

,-, fe] + c.

16 RESULTS OF EXERCISES

8. i esc 2 x + tan -= — | sec x cot x — § log (esc # + cot #) + C.

io. |io g ^ ana; 7} + c.

3 tan £ + 1

Page 217, §111

1. tV log (x - 2) + A log (x 2 + 2a: +5) + tt tan"* (^^) + &

2. log (a + l) 2 + g±± + C.

3. _ # fog * + f log (x - 3) + J ^^ + C.
4 v ^log|^ 2 + ftan-i| + a

5. f log (* - 1) - f log (x 2 - 2x + 5) + tan" 1 (^^) + C.

6. flog (x + 4) + ft log (2x + 1) - I l + C.

{Zx — 1)

7. log Vx 2 + 1 + tan" 1 s - 3(z 2 + 1) _1 + C.

8. 1(2 + l) 2 - | log x + V- log (a + 3) + V log (x - 3) + C.

Page 219, §112
1. lira 2 . 2. a 2 . 3. §7ra.

4. 10rr. 5. iwa\ 6. 49tf.

7. 47ra 2 . 8. 200 foot-pounds. 9. 300 pounds.

10. 6a. 12. Iwab. 13. Ixa 2 .

14. 8a. 15. &ra 2 b. 16. 11.08.

17.

\ira 2 sinh — - +
a

wax.

18.

37ra 2 .

19.

2.078.

20.

\ira z sinh

2b
a

+ i7ra 2 6.

21.

3h*a*.

22.

6f.

23.

72.

24.

2.

25.

-1].

^i 72
Clog^.

27.
29.

3tt.

28.

^ 1 + V-

a

Vl +a 2
a

30.

#7ra 2 .

31.

25tt.

32.

fTra 2 .

33.

4tt - 21|.

34.

9704 foot-pounds.

35.

9498.

36.

63,540 pounds.

37.

a 2 sinh- 1.

38. £ [2tt Vl + 4tt 2 + log (2tt + Vl + 4tt 2 ).

39,

RESULTS OF EXERCISES 17

a I y/l — e sin 2 u du, where e is the eccentricity of the ellipse,
Jo

u

= - — d, and a > b

40. (512) -(2340.4)*-.

41. ottW.

42. ,Va 3 (3ir - 4).

43. iwkr 2 h 2 .

44. 4690 foot-pounds.

45. ¥*•.

46. a sinh — .
a

47. 8a.

48. 27r 2 a 2 6.

49. 3-f T a 2 (207r

+ 1 - 39V3).

50. 184,300 foot-pounds. 51. -^a 3 .

52. y-rra 2 . 53. llx. 54. 2.278*- - 3\/o.

55. -V 6 a 2 . 56. hraK

Page 226, §113
1. 2. 2. No limit. 3. No limit. 4. 2 a/2.

5. No limit. 6. 2 y/a. 7. Jtt. 8. §x.

9. No limit. 10. 6. 11. 3(2* + 3 f ). 12. fxa 2 .

Page 227, §114
1. 1. 2. 1. 3. 1. 4. 2. 5. 47ra 2 .

Page 230, §116
1. a/29. 2. V40. 3. Vl3. 4. Vl3. 5. Vl30. 6. Vl4.

Page 231, §117

1. 1V3, iV3, |V3. 2. -^, ^|, -^.

o - 1 2 - 3 n

3 - 77^> TTtv 77?7- 4. jV3, jV3, *V3.

5.

Vl4' Vl4' Vl4'

- r - 7 4

V66' V66' a/66'

Page 232, §118

Q — 90

1 and 2. -^=. 2 and 3. —7=.

V87 V406

2

3 and 4.

V42*

Page 233, §119

1. x — -y/2y —2 = 4, if the normal extends into the eighth octant.

2. x - y + V22 + 10 = 0.

18 RESULTS OF EXERCISES

Page 234, §120

1. — 7=z 7=2/ - —7= z = —7=; — ■ f=; eighth.

V14 V14 V14 V14 V14

2. - §V3z - Wly - Wh = WS; iVS] seventh.

. 1 3,2 33",,

3. —7= a 7= y H —=_ z = —y=; -7=; fourth.

V14 V14 V14 Vl4 Vl4

1 *? R ft

6. — 7= x -\ 7= 2/ = — 7= J — 7^ J in the :n/-plane between the first and

V5 V5 V5 V5

fifth octants.

Page 236, §121

- 1 " q ' q ^ q *•• 7 ~ _ 7 ' 7 a.

*5 o *> 2^—3-4

Page 326, §122
1. 79° 7'. 2. 122° 19'.

Page 238, §124
1. 4VI4. 2. |V6. 3. HV38.

Page 240, §125
z-f y — # z -

2.

V26 V26 V26

z-f_2/-0_ g - |

V26 V26 a/26
x - _ y - _ z -

1

V35 V35 V35

Page 241, §126

2. ?/2 + g 2 = (3.2 _ a 2)2 #

Page 249, §130

1. (a) 0; ?f ; - - 2 -

2/ 3 2/ 2

2. (a) 2s. (6) — : L

RESULTS OF EXERCISES 19

Page 252, §132

1.

3.

2. i 3. - ^ 4. Irf.

5.

ia 2 (w - 2).

6. 15,750 7. fa 3 . 8. fa 1 - |a«

9.

_6_
35'

10. i 11. A- 12. |7ra 3 .
Page 255, §133

1.

•tV

3. Aa 2 (3* - 8). 4. T Va 2 (37r - 8)

6.

3i

8. 0.879.

Page 258, §134

1.

2oV / 27r.

2. i 3. V-a 3 - 4. T V.
Page 260, §135

1.

9 T

2. 37ra 2 . 3. a 2 . 4. wa 2 .
Page 262, §136

1.

¥^ 3 .

2. \mr z . 3. |aoc.

4.

ra /•(«§_

'' dz dy dx.

Jo Jo

Jo

5.

87ra 3 .

6. }f. 7. iV27r.
Page 270, §140

1.

£ = y = A

2.

Part Li =

io « -■ s p + o ~ — 4a ~ _ a

21 ,y 12 . fart 3. x 5(3?r _ 8 y 2/ 3?r _ g

Part 6. x =

s f£, £ = tV

4.

- 4r

07T

7. x = y = — 11. x = Jr.

7T

12.

X = fk

13. 5 = fa.

14.

- (6/i -
X =

3

a) (4/i + a)'- + a 2

10 [(4fc +a)- - a-]

15. On a line joining the apex with the centroid of the boundary of
the base, at a point two-thirds of the way from the apex to the
centroid.

16. x = -n-a, y = . '/. 17. x = ira, y = in.
io - - 256a -

18 - x = 1J = 31&T 19 - £ = IJ = '"■

23. z = y /t7to, x = fa.

20 RESULTS OF EXERCISES

Page 275, §141

— — An
1. 47r 2 a&. 2. 2ir 2 a 2 b. 4. x = y = J?"

Sir

Page 276, §142

1. Part 1. x = %a } y — 0. Part 2. x = %a, y = -^ —

rt - A 76 v 2a rt 2a sin a

2. x = f a 777^ 3.

4. x = y = |

105 7T 3a

a

Page 282, §145

1. \ab(a 2 + 6 2 ); T Va6(a 2 + b 2 )) £a& 3 ; ^afc 3 .

2. T Va6 3 ; 3Va6 3 . 3. #7ra 4 . 4. ixa 4 . 5. §7ra 4 .

6. A. 7. i P L 3 ; t VpL 3 . 8. ifcM 10. JraL 2 sin 2 a.

11. 2ar 3 ; fr 3 (2a - sin 2a); §r 3 (2a + sin 2a).

12. \bh\ 13. ^bh\
14. i7ra6 3 ; i7ra 3 &; i*-a&(a 2 + ¥).

Page 283, §146

1 35^4. 2 Va 4 O a 4 (457T - 128)

1. T6 -7ra , 3^a . 2. 192Q

3. |a 4 (2a - sin 2a). 4. Ja 3 (2a - sin 2a).
5. |aa 4 ; \ol& — %

Page 287, §147

irb 2 h{Ab 2 + h 9 ) , irb 2 h{W + 2fr 2 )
80 ; 60

2. =§ (3r 2 + 4/* 2 ); ^ (3r 2 + &*>.

3. j^abcic 2 + & 2 ). 4. ±7rr 4 /*.

5. MR 4 ~ r*)h; ^ (R 2 - r 2 ) (SR 2 + 3r 2 + Ah 2 );

j| (£ 2 - r 2 ) (3# 2 + 3r 2 + A 2 ).

11. ^irr% 12. A^^(3a 2 +L 2 ).

14. iV3a 4 ; |\/3a 4 .
Page 292, §151

2. _ 6 3 .

V#[4 + 9a;] I

8.

t 8 5tt(# 5 - r 5 ).

13.

T V7ra 4 .

1.

2

[5 -&c + 4z 2 ]f

RESULTS OF EXERCISES 21

3. 2x% 4, 2(1 ~ 3x)

{x* + 1]J [1 + (2s - 3s 2 ) 2 ]!

. 18s° ■>

0. , D.

[x« + 36]^ [4s + l]i

7 2x| 30z 4

9.

3[xi + 1]! " [4z 5 + 9]^

6s 2
[4z 3 + l]t

Page 293, §152

1 o 1 t

• 2* — ■ CSC — '

" a[sinh 2 1 + cosh 2 t]i * 4a ' 2

Page 296, §152

1. 216x 2 = (Sy - 3) 3 .

Page 299, §155

1. y 2 = 4px. 2. x 2 + y 2 = a 2 .

3. x 2 — 4ty 2 = 0, two straight lines.

2 2 2

4. x 3 + y 3 = a 3 j where a is the constant length and the coordinate
axes are the mutually perpendicular lines.

5. x 2 = 4?/. 6. y 2 = 4p(x + p).

Page 313, §163

/>»2 />«4 /*»6

l.coss = l- | - I -+ | - I - | - I + . . .

(# — a )2

2. cos x = cos a — (x — a) sin a — - — r-= — cos a

'_

, (x - a) 3 . ,

H r^ — sin a + . . .

h 2 h z

3. cos (a + h) — cos a — h sin a — ,-r- cos + 1-5- sin a + . . .

I ^ [_£_

4. sin (a + h) = sin a + h cos a — nr sin a — r«- cos a + . . .

«_ LA

5. Fart 1. — . cos x 6 ; Part 2. — . sin z 3 ;

/i 3 /i 3

Part 3. . _- sin :r 3 ; Part 4. — . cos x 3 .

x 2 . x 3 . x 4

,.«..! + , + -+-+£.+

22

RESULTS OF EXERCISES

e a

e a

1 + (x - a) +

7. e

8. 6

9. log (1 + x) — x -
10. log (1 - x) = - \

(a; — a) 2 (x — a)

12

+

13.

+

]•

a^ a) d

2" + "3 ~ ' * '

* + T + T +

]•

11. tan -1 a? = x

x 6 x*

3 + 5

+

12.
13.

1.

4.

7.
10.
12.
14.

(a) 1.39561; (b) 1.105171; (c) 0.052336; (d) 0.5403.

0.8387.

Convergent.
Divergent.
Convergent.
For all values.
For all values.
- 1 < x < + 1.

- 2 + 2i.
1.

14. 0.5299.
Page 319, §165
2. Divergent. 3. Divergent.

5. Convergent. 6. Convergent.

8. Convergent. 9. Convergent.

11. For all values.

13. - 1 < x < + 1.

2. -* +

Page 323, §169

^*

2 '*

5. 1.

Page 324, §169

3. - 1.
6. - 125.

9-n-i

2. V2e8, and V2e 8

1.

5.

9.
13.
17.
21.
25.

12.
13.
14.
15.

1.

n.

co ? if n > 0.

1.

Page 327, F §170

2. i 3. 0.

6. f. 7. 3.

10. f. 11. 0.

14. 0. 15. oo.

18. 1. 19. J.

22. 1. 23. 1.

4. 3.

8. - 6.

12. 0.

16. oo.

20. 1.

24. 1.

Page 331, §171

372 cubic inches per hour.

— 8.4 pounds per square foot.

— 25.4 pounds per square foot per second.
5.32 cubic feet per second.

RESULTS OF EXERCISES 23

Page 336, §173
1. x 3 y 2 = C. 2. sin - = C. 3. Not exact.

y

4. e x » = C. 5. Not exact. G. —„ + x - y 2 = C.

yc

7. Not exact.

Page 340, §178
2. cos y = C cos x.

3. [x + vtt^] [2/ + vTT7 2 ] = c vtt^.

4. 5x 2 y = Cy -2.

5. a; + y + 4 = C(x2/ + 2a; + 2 27 + 3).

7. log (z?,) = x — y + C.

8 -'-V

= e x + C.

9. log (xy) + \ - 1 ■ = C.
x y

10. log {x 2 y)

= y + C.

11. sin 2/ = C(l - e*) 3 .

Page 242, §179

1. log y = ^ + C. 2. 32/ 3 log 2/ = x 3 + C?/ 3

x

a;

3. (y + 2x) 3 (y + x) 2 = C. 4. log y + ^ = C.

5. y + Va^TT 2 = ^ 2# 6# log x + sin 1 = c

7. sin" 1 V r = log x + C. 8. log 2/ + - = C.

x 2/

Page 344, §180

1. ye x2 f= x + C. 2. 2/ = 2(sin a; - 1) + Ce~ 6in x .

3. ?/ = tan x - 1 + Ce" tan *. 4. 3?/(l + a, 2 ) = 4x 3 + C.

5. 62/(1 + x) 2 = (1 + z) 6 + C. 6. y - az + CVl - x 2 .

1

7. ?y = x"(e* + C). 9. 2/ = Cx 2 e* -f a; 2 .

Page 345, §181

1. sr« = |s 3 + Cx 5 . 2. y~ 2 = x + J + Ce 2x .

3. 49?/ 3 T 7(x + 1) + 1 = Ce lx . 4. t/" 1 + a = CVl - x 2 .

6. y~ * + ?z 3 = Cxi 6. I/" 1 = 1 + log x + Cx.

7. y*(x + l) 6 = ^x 10 + 2x 9 + 4 8 ^x 8 + 6 7 °^ 7 + l i~x* + V* 5 + i* 4 + C.

24

RESULTS OF EXERCISES

Page 349, §183
1. y = Cl e" + c 2 e->*. 2 . y = C x #* + C 2 e~*.

3. y = C l6 * + C*'*. 4 . j, - C l6 ». + C 2 e~^

5. 2/ = Cie 7 * + C 2 .

I. V = e 2 *(Ci + (7 2 z).
3. 2/ = Ci + C 2 z.

1. 2/ = •"*■ [

Page 350, §184

2. y = e 4 *(d + Caa?).

Page 351, §185

A cos

— 2~ a; + # sm — — - £ J

2. ?/ = A cos 3z + £ sin 3*.
4. = A cos o>2 + .6 sin ut.

3. 2/ = A cos 3 + £ sin x.

/-~

LIBRARY OF CONGRESS

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003 527 321 9 *

LIBRARY OF CONGRESS

003 527 321 9

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