245

488 Rozwiązania i odpowiedzi

17.122. / = -(-^3 + ^-₂ +    ) n'3-2x+x²

\9x³ 54x² 54xj

27^3

ln

3 — x — \/3\/3 — 2x + x²

17.123. I = L ¹ ^,+„_/² - )Vl-4x + x². \9 (x-2)² 27 (x —2)/

DO ROZDZIAŁU XVIII

18.30. / = ^sin 12x+|sin2x(‘). 18.32. / = y₅sin5x + jsin.v.

18.34. /= — 4rCos6x—ysin2x.

12    4-

18.36. / = |-sin 4x + ysin 2x.

18.38. / = isin3or-TⁱTsin7x.

O    14-

18.40. /= —jCosxsinx(sin²x+y)+|x.

18.42. I = isin⁵x-ysin³x + sinx.

18.44.    cosx#0; 1 = 1 tg⁴x-|tg² x-ln cosx.

18.45.    sinx#0; /=-yCtg³x+ctgx + x.

18.46.    sinx^0; / = -yctg⁵x+yctg³x — ctgx — x.

18.47.    /=yCOS^?X —icos⁵x.

18.48.    / = *jy COS^{1 3} X — yj- COS x+yCOS⁹x-jcos⁷ X. sin⁴x 4sin²x 8\cosx

I = — -jy cos 5x-icosx.

I = -yCOS4x+-jCOS2x.

/ = |sin 3x —jysin 7x.

/ = isin2x-2sin4x.

I =\cos³ x — cosx.

18.31.

18.33.

18.35.

18.37.

18.39.

18.41. / = yC0SXSinx(C0S²X + |)+|x. 18.43.

/= —|cos⁵x+|cos³x—cosx.

18.49. /= sin x —

15

15

/sin x smx\    x

18.50./= - cosx+—.    18.51.

4    8/    8

/=-

sin⁴x sin⁶x

sin⁵x 2sin⁷x sin⁹x 18.52. / =-+-

5    7    9

18.54. cosx#0, / = ln|tg(-jx + /x)|-sinx.    18.55.

18.56. / = - 2 ^/(l +2cosx)².    18.57.

18.53. sin x t^O, / =

/ = 3\/sinx.

- 1

7 sin⁷ x

18.58. / = ln11 + sin² x|.

1 = — 2 V1 +cos² x. 18.59. / = arcsin(sin²x).

O W rozwiązaniach pomijamy stalą C.

Do rozdziału XVIII

489

18.60. / = -

1 + sin² *

18.61. / = sin x—cos x.

sin x

+ i|n|tg±*|.

cos*

18.62. sin*=^0; /=--

2sm *

18.63. cos*#0; 1 =-=—+iln|tg(j7t+i*)|.

2 cos²* ²¹⁴    21

COS X

18.64. sin*/0; 1 =-=--fctg*.

3sin³*

18.65. cos*#0; / =

sin *    3 sin *

4 cos⁴* 8 cos²*

+f ln|tg(j7i+5*)| •

18.66. /= -

—-z—I--—z—H--rr" I cos * + Aln |tgi*|.

6sin⁶* 24sin⁴* 16sin²*/

18.67.

18.68.

18.69.

18.70.

18.71.

18.72.

18.73.

sin*#0 i cos*/0; /=--₅—l-lnltg*|

2 sin *

sin*^0 i cos*#0; / = 5tg²* + ln |tg*| .

sin*/0 i cos*#0; / =--₅---—;---r~r~

2 cos * sin * 4sm *

+ 3ln |tg.

1 =

—-3 l*ctg 2* .

3sin*cos *

sin*    , ,    ,    . .

i = sin* + --— +fln tg(i7t-j*) .

2 cos *

cos*#0 ; /= - isin³ *-sin* + ln |tg(-jTt+2*)| .

sin*#0; / = 4 sin² * — 21n Isin *1    —5— •

²    1    1 2 sin²*

18.74. cos*#0 ; / =

1

7 cos⁷* 5 cos⁵*

18.75. cos*^0; 1 =-5—hfln |tg(iit +4-*)l •

2 cos *

18.76.    /=4arctg(itg-Ł*).    18.77. i = ctg(^it+|*).

18.78. sin(jir+*)^0 ; 7 = 4^2 ln |tg(2-7t+3-*)| .

Wskazówka. Wykonać przekształcenie:

sin*+cos*=sin* + sin(iTt+*) = 2sin(-in+*)cos(-iit) = 72sin(in+*) .