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Biblioteczka Opracow ań Matematycznych

₌ _ O"    !n \x\__1_

4x⁴    8 .t⁴    32 x ⁴

62

J.rln

_ ^2lnM

i/ = In ^J |j| du=^£±dx

„2

,v' In ' |x|

d\> = xdx

u = ln|x| du = — x

,    x²

dv = x v = — 2

- J.t In xdx

x ‘ In ‘ .v

x"ln’x    a:* In x    | _r    x"ln*x    x* ln    x x‘

=-—--^Li + — I xdx =-C-l--— + — + C

7    7    7 J    7    7 A

dx

u = In i du -

x

, dx    - 1

dv = — v = -

X    X

64/

Jx In |x - 1 |c/r =

x’ In |x - 1

“ 2

65/

Jeos(lnx)& = = xcos(lnx)+

In |x I r dx

=--—+ J — =

r    ^J r

a = In |x - 11 du =

x — 1

X *

= X<&    V = -

2

In Lt |

——- —+ C

*² ln|x- l| _ ^ | x²dx _

2    2 Jx- 1 “

l_lr_{JC +} i ₊ _U_{& =}£lLⁿk-J!_£l. £.. ^In!^y-^|l _{+ r}

2 ^J    x - 1    2    4    2    2

/i \    . sin(lnxW*

u = cos(lnx) du =----——

dv = dx

v = x

. /,    \    . cos(lnx)

w = sin(ln x) du - —-—-

dv — dx

x v = x

= xcos(lnx)+ |sin(lnx)^ =

= xcos(ln x)+ xsin(ln x)- Jcos(lnx)dx

l Jcos(lnx)ir = xcos(lnx)+xsin(lnx)

r /, v, xcos(lnx) xsin(lnx) „ Jcos(In x)dx =-ⁱ-- +-²—L + _c

jarcsin xdx =

u = arcsm: du =

dx

= x arcsm

in x - J

xdr

l-x² =/²

xdr = -tdt H&L j arctgxdx =

dv = dx u = arctgx du =

= x arcsm x +

xarcsin x + Vl - x² + C dx

75“

dv = dx

1 + x^J

V = X

r xc

= xarctgx - J —

xdx

= xarctgx -

1 + x² - t 2xdx = dl xdx =

dl

1 _fd/

= xarctgx -- Jy =

= xarctgx - yln|l + x²|+C

68/

Jxorc/gx<ir =

u = arctgx du =

dr

l + x^J

2

x²arctgx 1 rx²dx x²arctgx

dv = xdv

■łf

_I f** ^{+ l}~'jr _ x²arctgx _J_ r^. ₊ J_ r dx _ x²arctgx x _[ arclgx _| ^ 2 ^J l + x²    2    2 J 2 Jl + x²    5    5    ?

dr

69/ |x²orcrgrdr =

2 2 u = arclgx du =

l + x² _r3

dv = x²dr v = —

3

2    2 2

x’arctgx    1 i- x'dx

-tJ

3 ^J1 + x

x’arctgx 1	f, | * Yj. ^'^gx	x^2	1	1 + x^2 = /
3 3 J	I^1' l + r-f' 3	6	3	2xdx = dt

x ³ arctgx x ²    1 r dt _ x³ arclgx x ² ln |l + x ' | ^ ^

~    r’ 6 '7"    3    6    6    ⁺

- 19-