09 (4)

Biblioteczka Opracowań Matematycznych

54/    |e*smxdx=

u =e^x du- e^xdx dv = sinxdx v = -cos.r

=-e co&x+

u=e^x du=e?d\ dv=cosdx v=sinx

= —e^x cosx + je¹ cos xdx = =-e‘ cos-r+e¹ sinx- je¹ sinx<ix

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

2 je^x sin xdx = e^x sin .v - e^x cos x

e sinjc-e cos*

je^x sin xdx =

55/

je~^2x sin 3xdx =

- e~^2x cos 3x 2

u = e~^2x    du = -2 e~^lxdx

,    . _    -cos3x

dv = sin 3 xdx v =-

-c cos3x 2

— je ^2x cos 3xdx

u = e ^lx du = -2e~^lxdx

,    , ,    sin 3x

dv = cos 3xdx v =-

- e~^2x cos 3.v

(e sin3x 2 r __u . ^{1 2}

-e^2lcos3x 2e²'sin3x 4

— fe'^2lsin3xaV o J

Stąd:    j^e *’^sin =

-e‘^2,cos 3x 2e~^J*sin 3x

je"²'sin 3xdx =

-3e~^J'cos 3x 2e'^2,sin 3;

13

13

56/

r. 2 x ,	X	dx — -dl		u=sin/	du=castdi
sin -dx= ^J 3	3~' dx=3dl	3	= 3 Jsin7a/ = 3 |sin/sin/^/ =	*v=sin/<*	v=-cos/

= -3 sin / cos/ + 3 Jcos²fc* = -3sin/cos/ + 3j(l -sin²/}* = -3sin/cos/ + 3 jc*-3 Jsin²/*/ 3 Jsin ²tdt = -3 sin / cos t + 3/ - 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 ^2 = — 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 ² 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-

1

-+—    \e * sin3mv

2

   3    J