10 (17)

Biblioteczka Opracowań Matematycznych

₌ _ (inj^iy ln |x|__1_

4*⁴    8.t⁴    32 ac⁴

+ C

62

Jat ln ² |jr(ir =

u = ln ² IacI du =

21" W

dx

.t² In ^: |x|    , .    ;t²ln².x

dv = xdx

u = ln|jc| du = —

.    x²

dv = ,v v = —

?

J    V -

■ - J.x ln xdx =

x' In^:IacI x^z ln|x| | _r jr' ln²|.tc| ot² Inljrl x²

--^L-¹--— -^L— \xdx =-—--■—- ^j—

^ 2

■ + -

2 2

+ — + C 4

dx

u = ln |j| du

. dx    - 1

OV = ——    V = -

X *    X

64/

J.xln|x- l|e£c =

651_

Jcos(lnx)a!x = = xcos(lnx)+

=    r*_ ₌ ^lnH 1 ! _c

r * r'    r    r

u = ln |x - 11 du =

dx

.X - 1

x²

dv = xdx v =-

2

lnj-r - 1[ _    | _x^ldx _

2    2 ■! x — I

₊ ¹ _dx = * ^lnF- ‘I * ■"!*- '1 , _c

x-l    2    42    2

/. _x , sin(lnx)oEx « = cos(lnxJ du =--¹-^£—

dv = dx

V = x

.    \    , cos(ln.x)

u = sin(ln.x) du = ——--

dv = dx

x v=x

= xcos(lnx)+ Jsin(lnjx)ć£r =

= .xcos(ln.x)+.xsin(ln.v)- |cos(ln.x)a!x

2 Jcos(lnx)ix = xcos(ln.x)+.xsin(ln.x)

r /,    xcos(lnx) .xsin(ln.v)

Jcos(lnx)ix =-^^-’- + 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-