249

496 Rozwiązania i odpowiedzi

20.56. L = r[if²]5 = i7c²r.

20.57. L=[|l³]5-|n^:

ir.-

20.58.    L= ~Yz^a [cos i V1 +cos²1 + 5_%/3 = ln|\^3cosi Wl + 3 cos²!

= |a(2+iV31n|2+y3|).

20.59. L = 2a    *' T^Hs/Ścosfi Wl +3cos² f,|^ •

20.60.    L = 40.

a (    1 , J2 + l\

20.61. L=-l+—pln^p- .

2 V    2^2 sll-\)

5 a

20.62. L=^—7= (2^/3+ln (2+^/3)).

20.66. K = jiln —.

a

20.67. V = lim nil-

a-* + oo \

20.68. V = \n², S = 2re(_>/2+ln(_>/2 + l)).    20.69. K = ^-81n2.

20.70.    V-ja\2-j2), S-w¹ (,/6 + I+i b *I±1Y

20.71.    V = ±n, S=|jiV2.

20.72.    V=lna³(e²+4-e~²), S=f™V+4-e^-2).

20.73.    4nr.

2r-x'~'²

20.74. V = n J~±x

³+rx²+4/-²x + 8r³ ln

= 2 + 3r + 4r²+8r³lr

> = 2jtr f

x — 4r

8 r

2 r , u

3x — 4r^_|2

2r—2j 2r—1 ’

r-2

.    V8rx—3x -i—arccos

\_x-2r    3    4 r

20.75. V=n[ — COSX + COS³ X — ^COS⁵ X + i COS⁷ X] o' = ‘®n.

20 76. V = 7r[sin x — sin³ x + ³ sin⁵ x — ^x- sin⁷ x]q ² = ^ *.

6/4

2ji    4n²a² . , 2n

20.77. S = 2it | a cos — x ■ /1 H--sm² — x dx =

b V b b

f2na . 2n 4n²a² . .2n

• I -cm - v /-citW -

= —-P

%n²a [

sin —- X b b

, sin² — x + 1 + b    b

+ ln

2na . 2n 4n²a² . , 2n —— sin — x + / ,, sm² — x + 1 b b V b²    b

= \ a Jb² + (2na)² + — ln ^{2 v}    4ji

6/4

2na 4n²a²    ,

-v ⁺ Jnr ⁺ '

tc[3_N/x² — 6x + 15 + 101n|x — 3 + ,/x² — 6x 4- 151] ² _

20.78.    V =

20.79.    V =

20.81.    V =

20.82.    S =

20.83.    S =

20.85.    V =

20.86.    V =

20.87.    V =

+ 101n| - 2 + >| - 3^/Ti - lOln| - 3 + ^/15|] .

8rc[x - Ix³]³ = 16ti.    20.80. V = 32rt[x - ix³]³ = 64tt.

T lt[4x —    1t.

2007i|arcsin |xj^ = 200tc², V = 50071^71^/25 — x² + arcsin^xj~³ = 5.

H<> ^{+ 3x)i}]; - i”'    ²⁰-^{84 s} " 5"[° ^{+ 36}*‘>⁵]i-|'/37-

r ‘Ti

L *-u₂ ³

K-fcE-H-

sirr^Sj-    mr.irf, I-Srf.

20.89. V 20.102. /„ = i pab³. 20.104. I_n = ±pah³

²"[i*⁴-5«*⁺ś>’]f -S -    + !<♦+ !<^!]f - 3«.

20.103. /„ = ' Tt8r⁴/7.

20.105.    = Iti8rfc².

20.106.    l_x = nkr³.

20.107.    7_X = | p [| x(5r² - 2x²) V r² - x² + fjr⁴ arcsin (x/r)]1 _r = i tcpr* .

20.108.    I_x = £ndr³ .    20.109. I_x = ±Ttdab* .

20.110. I_x = ±da³(e + e~^l)(e² + lO+e~²).    20.111. M_x = \pah².

oa²b²

20.112.    M_a = ^pa²b, M_b = \pa²b ,M_c=^-===.

yj a +b

20.113.    M_x = ±pah² .

20.114.    M_x = \p[x²Yo° = \x₀yl, M=yJYp[\^²Yo^a-\xly₀.

a²b "16* ’ ab²

'iii²

20.115.

ba² [2n    . 2ti 2tc 1

M_x =— —x+sm — xcos —x = 871 [_ fo    b    b Jo

ab² [2n    2k    2n T^/4

My= ~ 2 T~ * Sin ~7~ X +^COS —— X

47C^ 1_ ł>    b    b Jo