ANSWERS TO THE EXAMPLES
Examples 1.
i A/3 1 I I 1 V3 1 i A/3 1
2,— 2 ,-A/3' 1/2' V2> _ I ; 2' 2 , ~3;-2,- 2 , 1/3
5r 377 25r 7r
(v) nr+4.
- 8.(i) 7; (ii) 7; (iii) r; (iv) 2r; (v) ?T; (vi) --; (vii) 4'r;
222,4 (viii) — ; (ix) 27r; (x) 16- r .
- 9.(I) IS; (II) 22' (iii) g; (Iv) 52 ; (v) (3n —I) LT.
- 10.— cos 0; — sin 8; — tan 0; tan 0; cosec 0; — cosec 0; — cot 0; sin B; sin 0. 56. 13.1929240 356. 2
12. 65' 85' 13 A/5' 13 ./5' 289' 5' 33'II
- (4n + 1) or (2n + 3) 72 + tari—1
2I.34. (2n — i) 7r — a.35. --3
~rlo11 + ~5'n7rnn36, 7177 + (— I)n sin —i43 i • 4 or s .
- Solving in the ordinary way, x = 2. On substitution, however, x = 2 gives a positive value to the left-hand side of the equation, whereas the right-hand side is negative. Strictly speaking, therefore, there is no solution.
- 2,17r ± 2 , 4n7r ± r or - (4n7r ± 7r).40. x = 4 •
,/ 348. ± — . 2
2. -6' 4' 6 ' 6.
- (a) 6 ; (b) 3 ; (c) 6 ; (d) 4; (e) o.
- (a) 1-008; (b) 1.006; (c) 2.233.
7. (i) n7r + (— I)n4; (ii) 2nr ±7;(iii) nr + 5 6' (iv)r2nrr +4- ;46. {in + } r or {n — k} r.49. a=inr+(-1)n4r; $=:nn+(- 1)rz r•5o, sin {a+(n—1) R} sin in3 sin' 3
384 ACTUARIAL MATHEMATICS Examples 2.
1. 58. 2. 30, 42. 3. 15.
4. 1.9, 4'9. 5. 11 lo. 8. 6ah3.
9. A (— 11x3 + 252X2 — 1051X + 1344). 10. abcd. Io!.
- ab" (be-1); abex(b°-1)2; ab`x-(b°-I) be -z.
I)'o-I] -
- (i) Ix (x — ,) + k,(ii) cx/(c — I) + k,
(iii) 3X (x - 1) (x - 2) +(x - 1) + 3X + k, where k is a constant.2_3
- -(x+2)(x+3) (x+3)(x+4)'
46(x+ 2) (x+3)(x+4)+(xt 3)(x+4)(x+ 5)2
- x(x-1)(x-2)
16. (1) an!; (2) eax+b (ea — 1)n.18. 55.
- Ix (x — 1) (x — 2) (x — 3) + 2x (x — 1) (x — 2) + x (x — 1) + I2x + k.
- 2225.22. 20.23, — 161.24. 229.25. 1261.
- x(4) - 6x(3) + 13x(2) + x(') + 9; 4x(3) - 18x(2) + 26x(') + 1;
I2x(2) - 36x(') + 26; 24x(') - 36; 24.
- (i) (m + 1)m(m-1)...(m-n +2) am+' (b/a+x+m)(m-"+');
(ii) (— I)" (m + I) (m + z) ... (m + n) a-""" (b/a + x)(-m+"+')
- z cos (x + la) sin la ; sin a/ {cos (x + a) cos x} ; a - 2 sin (x + a) sin la.
- 6x; 6/(x + 1)2.42. y2 + 4ay = /32.43. 2 (x - 2)" - 2 (x - 3)".
15. -2or109.
19. a2x
+ (a2 + I)2 a4x.
15.
1. 465.
4. 182; 343. 7. 128.
10. 97,357. 13. 69,215. 17. 14.73658. 20. 5479.
22. 23. 24. 26. 29.
31. 32.
35.
Examples 3.
441; 653. 5414.
94; 396; 66z. 844; 746•
3. 300.
6. 89,920; 89,073. 9. 194'3; 279'9. 12. '98127.
'432; -.338; -.196. 19. 5281; 6504.
2'37223. 3.708; 3.711.
21. 2153; 1705.
2459; 2424; 2359; 2268; 2153; 2018; 1868; 1705; 1534; 1357; '017; '035; '052; •070; '087; '104; '122; •139; '157.
23'1234; 23'2039; 23.2914; 23'3865; 23.4898; 23.6019; 23'7234.
I•000. 27. •020660; '020625; '020628. 28. 58,844. 1; 2.10; 3'31; 4'64; 6'II; 7'73; 9'51; 11'47; 13.62; 15.97. 117.7; 114.2; II0.5; 106.7; Io2.7; 98'6; 94'3; 89'8; 85'2; 80'3; 75'4.
'24928. 33. Third degree: 275. 34. 459.
1[2=218; ui=o; ay=—19; ux=1876-1429x+36ox2-3ox3.
I180.
ANSWERS TO THE EXAMPLES Examples 4.
2. 47,983. 3. 2'8169. 4. 1.7243. 5.
1.
5745.
385 2300. 6. 460.
7 -1+m lm+mn+nl —lmn+mnp+npl+plm
12m2 ' 12m2n2 12m2n2p2
8. 13.18. 9. 14.942. 10. 20.43. 11. 162.
12. 659+224x+82x2— ;1.,x3. 13. 32. 16. I; 25.
18. 33 and 67 to the nearest integer. 19. 37.2.
20. 7.37. 21. 130,326.
Examples 5.
1. 33. 2. 6. 3. 47,692. 4. 3251. 5. 16.9216.
6. 2.85805; 2.86305; 2.86157; 2.86155. 7. 2017.
8. 3.5283. 9. 2196,2108,2022,1939; 1786, 1718, 1657, 1604.
10. •01625. 11. 3165. 12. 2290.1. 14. 4'034.
Examples 6.
1. 471.5; 2.7. 2. 13 3. 3. 2'019...; 2.018....
4. 43.1. 5. 8'34. 6. 2.751. 7. 16.9.
- 8.I.1576....9. 1.2134.10. 45.70.11. 1 85 .
12. 3.091.13. 3.667 per cent.14. 1.3713.15. 37.2.
Examples 7.
1. 4n (— we + 27n + 17).2. 1!2n (n + 1) (3n2 + 7n + 2).3. — 4195.4. 221 + 628.5. 3n+1 + 4 (n2 + 7n — 6). 6. A {2 (3n — 1) + 5n + An (n + I) (2n + I)).7 221:+1 — 2 — k (2k + I) (4k + I).8. A (n4 — Iona + 29n2 + Ion).
- 9.2n4 + i6na + 47n2 + 6on.10. 219 — 2095.
11. 4(n+3)(n+4)(n+5)(n+6)—90.12. in(n+ 1) (n + 4) (n + 5).13.{(3n — 2) (3n + I) (3n + 4) (3n + 7) + 56}.14. 1-1_n (n + I) (n + 2) (3n + 13).15. -11n (n + i) (n + 2) (6n2 + 57n + 137).16. -,.Al {(2n + 3) (2n + 5) (2n + 7) (2n + 9) (2n + II) — 1o395}.
n n(5n+ _13)n(n+ i)4(n+4)18. 12(n+2)(n+3)'19. 6(n+3)(n+4)•F25
17.
386 ACTUARIAL MATHEMATICS
n(3n+5) n(5n+ II)
20. 8(3n+ 1)(3n+4)' 21. 4(n+ 1)(n+2),
22. H n (n + I) (n + 2) (3n2 + 36n + Ioi). 23, 19 – 12n2 + 33n + 19 19
168. 6 (3n + I)(3n + 4) (3n + 7)' 168'
- (n + 1) (3n2 + 31n + 74).
+(n+3)(n+2)(n+ 1) n; in(n – I)(n–2)(n–3).26.— r — n.27. - - az (x2 —a — tax I + (aa (a + I)l + C.3a11-1)2
- I — 3 (n + I) ! + (n + 2) !.
12 {(3n – I) (3n + 2) (3n + 5) (3n + 8) (3n + 11) + 88o}.
- n (6n3 + 16n2 + 9n – 4).31. 2? n (7n3 – 34n2 + 89n – 254).
- 12n (n + I) (9n2 + 17n + 4); {21}1 (n3 – 3n2 + Ion – 14)) + 28.
- 3 + 4n 3 +1 {n +_ z} .35. 768'36. 34 – (4n2 + Ian + 17)/2n--1.
37. (x – 2) 3x.38.(3" – 1) + 1'_ (42n + 17n2 + n4).39. 42.40.(n+2)xn — 2 x"t+1—x x — I(x (x — I)2 'I 39 36n + 39 5410 (3n+z)(371 +5)1•I2x3 + 36x2 + 28x _+ 3 12x (x + 1) (x + 2) (x + 3) '43. 23.44. n22n+1.45. z — (n + I) (n + 2) xr` +2X 2 {2 — (n + 2) xn} + -2.x2 3 (I — x").I—x(I—x)(1—x)
C – 2x-2 (x – 11 (2x– I)!'
- +$1-6(2n– 1)(zn+ 1)(2n+3)(24n+54n+25).
ax' – (la – 3b)x3+(ea– lb+ c)x2+(tb– is+d)x+K.(I — x)r+1Examples 8.2. 20'796875; – u i •10. 5'254.13. 13'094.14. 5.319.15. 3.9634.16. 10.389; 10'475. Examples 9.1. 4 an-Q.3.(log a)2.5. 1 (a — k).7. o.8. 2a2.9. 2 9 3.10. I.11.12. e (log a —log b)13. n .14. I.15.1
2a16. oo .17. o.20. 4.46.
387
ANSWERS TO THE EXAMPLES
Examples 10.
z
5. i/a2 — x2 — x ; ex log x (log x + 1). v'az — x2
6. 5X4; an (ax + b)'°-1; xx (I + log x).
n
- 7.bn (a + bx)"-1; na1x log a; zx/n (a2 + x2) n ; x - log a.
8 2 (xa — I).x2; xn—1 (I — x)n—I (m — m + nx) ; x'—1 ex (x + m) ;x1-1 (I + n log x).9' — xlog aloxe x I x (1 + 2 log x); Io10x (loge Io)2 lox; Ilog.
lo;
- 2 sin x cos x; 2 cos 2x; — 3 cost x sin x; x sec x (2 + x tan x).
- 2X; sin x + x cos x; sect x tan 2x + 2 sect 2x tan x.
"I — xa2 VX sin xxx212. cos' x- ; 2x tan—1 x + -1 + x2; VI—x2cos x (tan—1 x)2 + 2 sin x tan l x I + x2
- (5 + 4x) I°g {x log (5 + 4X) + 5+ 4x log xl
—mx'°-11~ I - x2 {I - V I - x2}m'' x log x
- ax log a + ax°-1; xx (I + log x) - 2mx (I - x2)"°-1;
xxn xri-1 (n log x + I).
- 3 (I —4x2). 3 vx Vcos-I x — x.-x/
(1 + x2) (I + 16x2)' 22 V I - x2 V cos 1 x.y2II18. x - xylogx'19. x ex- 1'21. (i)xa - 2azxz + 4a4 ; (ii)nal(x2 - a2)= (x2 - 4a2)}'2x (a + bxn)}.
- - -4x jtan_1 2xz + a I-I
4x2 + I (2ax2 - J
- (i) ex xer {log x + x) ; (ii) {log 1 + x1 + x} if I -
26. 3 (log x)2 x flog x)z.l27. 1.2 (I — t2).— 4t8t(I + t2)2
23. (1 + t2)2 ' (I + t2)2' (I - t2)3 ' (I + t2)
20. feet per second.
25-2
388 29.
ACTUARIAL MATHEMATICS
2ex
32.
ve2x — a2.
34. (i) cot x; (ii) 4 + x2'
a sin a/x — cos a/x 2x2 ' I + sin a/x cos a/x
2x4 + y2
x3
39. (i) exxxx (I + log x); (ii) exsxx (I log x); (iii) exz.
(i) 62 (ad — bc) (ii) I I — 6 log x
(cx + d)4' x4
44. 4x2 — 4x. 46. 2 log a • 50. o. a
51. n l pa+q i pb+q
(— I) n. ~(a — b) (a — c) (x — a)n+i + (b — c) (b — a) • (x
1
pc + q
- I
+(c—a)((c—b)(x—c)n+i}
(— I)n n! !I - 6 — --- 2
l(x — z)n+1 (x — I)n+1
(— i)nn! {(c—a)(c — b)_(d—a)(d—b)
c — d (x — c)n+i (x — d)n+1 }
8 256
54. (I — x)5 + (I + 2x)5
r
57. (i) (— I)',-1 ?(n-- 3)!. (ii) (— I)n n!}I 5.3f + - I
xn—2 ' l(3x + )n+1 (x — )n}11•
59. ari}2x2e"x.
(n + I)!
(— I)n (b n a)2 ((x —Ib)n+1— (x — a)n+1} — (— I)n (b — a) (x — a)n+2•
70. p = (log s)2 ; q = log c log g (log cs2) ; r = (log g log c)2.
—cos0 r
(i) N'cos 20—cosec2 B cos2 28 (ii) 2 (I +xx2
3. 74. (o, o) and (za, — 4a3/3b2).
(i) (— 2, 3) (— 2, — I); (ii) (I, I) (— 5, I).
Examples 11.
x x2 x4 5
2. log2+-+--- ...
3• I + x z — 2+
2 8 192
6
x x2 x3 x2 x3 x' x5 x6
6. I + — + .... 7. I + 2x + - + + + - + -
- 1 2 24 ... .
2 6 12 40 720
I 3a2x2 + 2ax3 + x4
30.
33. 35. 37.
3 (a + x) (a2 + x2) (a2 + ax + x2)
I I+ VI—x4
2x2 loge I o' x6 — -- .
I — 2Xy
x2 + 3y2 — I
n!
ar = — (n — r)!'
(i) y ; (ii) 2y — y .
36.
38
41.
52. 53.
30.
71.
73. 75.
ANSWERS TO THE EXAMPLES 389 5
9. x—x +x —x .... 10. -+x+x5-~ .... 12. -4b.
3 5 7 4 2 4 22 3
13. +3a 81a3.... 14. x—ix3+aix5.... 17. -4k.
18. 1 + i x + ~Lx2 + ] Ax3 .... 20. I = - + x3 - x5 ex + 1 2 4 48 480....
22. px–sp(p2+6)x3+,:',-,p(p4+2op2+6o)x5....
Examples 12.
1. Max. 1; Min. — i i . 2. Max. 9a3 ; Min. — 3
3. Max. 3 ; Min. A. 4. Max. r, 2 ; ; Min. o.
5. Max. 73; Min. 69, 69. 6. Max. 63,63; Min. —
- 7.Max. 13; Min. — lo; Point of inflexion where x = o.
- 8.Max. 2 V ab + a + b min. 2 V ab — a — b9. Max. ,74T; Min. o.
2'lab—a—b'2 'lab+a+b.10. Max. — 2c 'dab; Min. 2c Va.11. an'112. Max. 41; Min. 3.13. zo7.8.14. e e .16. 3—3loge 3.17. a=—6; $=9.
- 125 yards from A along AB and then across the grass to C.
- 20. n parts.21. * {(a + b) — Va2 — ab + b2}.
63(a* + b3)2. Max.c'lab + h;— 'ab — h'aV~2 - 132. 165 feet.33. y = 2x3 +.34. Max. z ; Min. . x35. Max. — 6; Min. 2.36. Max. 12; Min. 8, — zoo.
38. x = o. 39. ± --i=. 40. ±
J2 1~3
43. Max. R; Min. o. 44. Max. 4.317; Min. 4.183.
45. Max. V; Min. 46. — 5. 47. (a + b)2.
22. 24.
27.
30. 2 'lab. 31. f (a + b)2.
23. 66 minutes.
390 ACTUARIAL MATHEMATICS
Examples 13.
1. o. 2. o. 3. 1. 4.
log, a.
7. — I. 8. /?nma3(m—n).
11. 1. 12.
9I
m 6 _I
n V za
10. r.
13. loge a. 14. I re 15. I. 16. a',. 24
17. s, 18. b+alogb. 20. oo. 2b—a a
21. a = 120; b = 6o; c = 180. 23. o. 24. o.
2
25. 3U. ' 26. _}u. 28. — x + y 30. u — uy.
y2 + x
33. .02455; — '0003; 0. 34. • 109. 36. keax+Ie:,
Examples 14.
Note. In the answers to questions on indefinite integrals, the presence of the constant of integration is to be inferred.
2.-n+1
.
1e2x
n+1' 3(—n+I)'
- 3.ix5 + 3x3a2 + xa4; ax + bx2 + 3cx3; — } (I + x)—2.
axaxax
- 4.— cosx; sinx; — cotx.5. loge a; loge a + bx; loge a + hbx2 + cx.
- 6.— i cos 2x; I sin 3x; i tan 4X.
- 7.l01 . — _I
g (x + I); 1 – x'12 ' (a + 3x)4'2 _bc
- 8.31ogx+x'(n— 3a) xn-2+(n— 2)xn 2+(n
- 9.ilogx+I; logx — tan' x.
- 10.a sin—1 x; a sin 1 ax.11.sin 3x + - sin x.
I2I1
- — (n — 3) yn_3 (n — 2) yn—2 + (n — I) yn—1J'
2L(n – 3) (I + x)n1-3 (n – z) (I + x)n-2 + (n – I) (I + x)n-l] .
- m
l l i+m y .14. fo I dxx215. log f ''Vx2 — 2xl . t x– I r38. 48; 24.
ANSWERS TO THE EXAMPLES 391
1+x
16. – log (I – x) – tan' x. 17. A log - -
I – x 2(I+x)
23. ex+ax0+bx+c. 25. IIa
6-*
26. 1x = ke—Ax—BeIloge c 27. A log I + t. 28. log I I x4. Examples 15.
Note. In the answers to questions on indefinite integrals, the presence of the constant of integration is to be inferred.
I I 2 I I I
1' n — I (I + x)n—' + n — 2 (I + x)R—2 n - 3 . (I + x)n—3' 2 ~x + - ,
(x— I)(x+2)i.
2. 2 log (x - 3) - log (x - I); 3X + I I log (x - 2) - 2 log (x - I); 2Vx—1 3 2
35 (5x + 6x + 8x + 16).
. 1 10x0-a0' — + 1'x
3
gxz+a2' x - ag
4az x+3
- 4.} log (3 + 4 sin x) ; log (4x0 + 3) ; log (ex + cos x).
- 5.sin-' 2x +log (x++ V2+x+x2).
3
- 6.– 3 log {A + } cos x}; log I (+ x +) xz + V- tan 1 2x+ 1
1-x33/x0 - x + I Ix3 - a37, log \/x-+x+ I' 6a31ogx3+a3
- V1 + x2; j log (4x0 + 3) + -2 - tan +' 21
V3V3
- A {cos x - k cos 5x} ; sin Ax + + sin
n —
- 3 (x2 - 2a2) 'ct0 + x2 ; I log - I -± x1
n' 1 + xn +Icx+b - kIcx+b/2+x2
- zklogcx+b+k or ktan-1 k12. -2 \/2
x2
- (i) Vx2 - 1 ; (ii) {x 1/xz - + log (x + Vx' - I));
(iii) (x2 — I); + Vx2 — I.
- — 4 log (x — I) — 2 (x 5- 1) — 2 (xI 1)2 tan-' x + I log (I + x2).
ax
- log t; t; log t t2.16. Ax' log x – ix2; a3 (2 – 2ax + a2x2).
17. ex (x2 — 2x + 2);(x3 log x — Ax').
392 ACTUARIAL MATHEMATICS
- sin x — x cos x ; x' sin x + 3x2 cos x — 6x sin x — 6 cos x.
{(x2 + I) tan' x — x}; ex/(x + I).eax (a cos bx + b sin bx)/(a2 + b2).
- -i log (2x + V4x2 — 7).22. log tan (far + ix).
23. — {log (I — x)}2.24. log x {log (log x) — I}.25, x (3a2 + 2x2)26 (x2 + zbx + c)"`}I3a4 (a' + x2)T2 (il + I)
- ~_{log (6x + I) + 2 V3 /3x2 + x + 8}.
V3
- x — loge (I + cx)/loae c.
log Vx2 + x + I + _ tan-I {(2x + I)/ V3}.V3
- V5 + zx + x2 — log (I + x + V5 + zx + x2).
- log x — } log (I + x2).32.tan-I (k tan ix).
33. I log x—'~2+VI+x2+I34. —sini3—XV2 x + V2 ++ x2 +V32x35. — I_ log{x+4+2'12 2+x—+x2}1 log x.N'212Isec= x36.log237. a sin-I (x/a) — Va2 — x2.2(a—b)a + b tanx38. Q {(x + a)i — x3}.39. .?_, {3x — 4 log (4 cos x + 3 sin x)}.
3aI( V a40. tan-I tan x}Vat+abIVa+b
- (a) y (log y — I) ; (b) y cosy — V I — y2.
- — I (2x2 + I)/(x2 + I)2 + a constant.45. ?} tans x2.
2(~46. log (x + Vx2 — I) — sec—I x.47. a2 _ b2 tan-I {V aa —+ bb tan ix}.48. — I (a2 + x2)- (2a2 + 3x2).49. ea' (x/q — I/q2).Jx3—2m1(2 — 2m) a2 (x2 — a2)m—I (2 — zm) a= Itn—1x _x—a za2 x2 — a2 4a3 log x + a'51. ix4 (log x)2 — 8x4 log x + 3I_ x4.
- — 2 {a (a — x) 1/tax — x' + a' sin-I (a x)/a} — (tax — x2)!.
Isin (x — a)
- sin (a — b) log sin (x — b)
50.
ANSWERS TO THE EXAMPLES 393
55. x3 sin 1 x + (2 + x2) I