09 (4)

Biblioteczka Opracowań Matematycznych

54/    ^e^xs\nxdx =

u =e^x du = e^xdx dv = sin    v = -cosx

—~e cosx+

u=e^x du-e^xd\ dv=cosdx v=sinx

= -e' cosx+ je¹ cosxdx = =-e^,cosjr+e'sin.t- je‘s\nxdx

A zatem biorąc pod uwagę początek i koniec obliczeń mamy: je^x sin xdx = -e^x cos x + e^x sin x - je^x sin xdx

21e^x sin xdx = e^x sin x - e^x cos x

je^x sin xdx = ■

e^x sin x-e^x cos*

55/

je~^lx sin 3xdx =

u = e

du = -2 e dx

dv = sin 3xdx v =

- cos 3.t

-e'^Jxcos3x 2 _f ,,

-1--—Je " cos3xdx

- e ^2x cos 3x 2

u - e ^2x du = -2e~^2xdx ,    _    .    sin 3x

dv = cos 3 xdx v = -

- e ^2x cos 3.v

l(e^2xsin3x    2 r    _u . - .'i    -e'^2xcos3x    2e^lxsin3x    4 r    ,, .

-- ---+- e -^xsin3*dEr =---------e'^2xsin3.r^v

3^    3    3^J    J 3    9    9 ^J

Stąd:    ^-Je'^j,sin ^3xdx =

cos 3 x 2 e ² * sin 3 jr

je ~^lx sin 3xdx

3    9

-3e“^2xcos 3x 2e~^Jxsin 3.t

56/

13

13

[sin²-<Zx= ^J 3

X	dx — -dt		«=sin/	du = costdi
3~‘ dx-3dt	3	=3 Jsm fcń=3 Jsinr sin/rff =	dv=sintdt	v=-cos/

= -3sin / cosr + 3 Jcos²/c// = -3sinfcos/ + 3 j(l — sin²/)* = -3sinłcos/ + 3 j*A-3 jsin²tó!r 3 Jsin ²tdt = -3 sin t cos t + 3t - 3 Jsin ²tdt

r . i.,. -3sin/cos/		3/ r	— sin cos . j x 3 3 x >in — =---—+ —
J31II iUi —	6	-- J 6	3	2 6
57/ fcos ^3 = — j 4	X 4 " ' dx = 4 dt	= 4Jcos^2 idi =	U = cos / dv = cos tdt	du = - sin tdt v = sin /

4(cos/sin/ + Jsin²/cfr)= 4cos/sin/ + 4 J(1 -cos't)di = 4cos/sin/ + 4 fdr - 4 jcołtdt 8 Jcos ² tdt = At + 4 cos t sin t

x . x cos —sin ¹    4_4

+ C

r 2 . i cos łsin t _

cos ' tdt = —+-+ C - —

J    2    2

58/

J %/x ln xdx =

u = ln x du = — x

i    ₂ -

dv = x²dx v = —x²

= f ** ^toW-f    = f ** H»|-| J***

3 ³ x

= yx^ln|x|~x^ + C

u = ln Lv    =

dx

59/

= x ln I jc

x In U - x + C

dv = dx

v = x

fdx =2

| ln |x| In |x|dr =	u = In |x | du - X
	dv = ln |x |dx v = x ln |x | — x

= In |jt|(.v ln |x| - ar)- JIn |jr(cat»r + Jdx = x ln ²|x|- 2x In |x| + 2x + C

61/

I

u = ln ‘ 1x1 du

2 ln |x|<ćr

dv = x dx

x

- 1 4x⁴

(H*!)' , i f^lni^jl^tżr

4x⁴    2 J x^s

u = ln |x I    =

dv = x‘dx v =

x

- 1 4x⁴

₌ (^ln kiy In kl i t±_ ₌

4i‘ 8x¹    8 ^J x⁵

- 17-