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Biblioteczka Opracowań Matematycznych

164/

(xarctgxdx

^J

u = arclgx . xdx

ch,' = --— V

du =

dx

l + x-

arctgx 1 r dx

2(l ₊ ^⁺2^f

arcigx

2(l-

r 1 ( x

+—arctgx 1 =

T~2

(x^ł - l)arctgx + x

4c^J ₊ l)

+ C

Do obliczenia całki 164/ wykorzystano wzór rekurencyjny (1.28). 165/

rarcsin*

u = arcsin* du =

dx

dx =

i-hr

w 1-

.rvl-.v

■ -ln

,    dir

X

1    1

- = /    x=-

X    t

- arcsin .t r dx

—:—^łJ:

arcsin*

x-K

+ 1

' .1

‘ .2

,    __r_T-c_ ,[-z£.,-iJ,₊V7q.

4-* *-4 ^JTJi-T 7^ ^    ¹

X-    t t\ i¹ t

rlf

i=^~

1 +c

arcsm x . arcsin x

- ln

„ arcsm x , + C =---ln

1+

+c

166/

rarcsin e’ .

arcsin e

		dt
e^1 =t e‘dx = dt	f arcsin tdt	u = arcsin/ du = —7== VI-/^2
dx=*	J t^2	, dt I dv = — v = --
1		t t

arcsin t f <A

+ /

/=j^-7===

\_

— — z t

1

t=~

-*ldĄ

,2 ^

= -In-

7⁺i?

/ = -ln

= -ln|lWl-eH+lne‘ = x-ln|lWl-<?H

Ostatecznie otrzymujemy:

f arcsin e* ±    . _r , , r.-1

J---dx - -e arcsine +x-lnl + vl-e

167/

^X^>arctgxd&

+ C

dx

u=arclgx du=

l+x

dv=ćdx v= —

₌xWc&_\_ r^ ₌ ^W_i

4 4^Jl+x² 4    4 \    1+^r

x*arctgx x' x arctgx x⁴ -1 )arctgx -x³ + x

4

168/

4    4

dx

+ C

1 rdu 1

r    ax    2dx i tau \    i

h\+4x²\arclelxY ‘"^C'⁸    “^_T+4? “ 2 V ~ 2u~ 2arctg2x

169/

+ C

f_^dv	u =arccosx jl &
^jV1-at^2 arccos^:x	du =—

(• du _ 1    1

' u² u arccosx

Jln(x + Vj' + l^t =

u = ln(x + Vx^J + 1) du-—fi==

dv = dx    V = X

+ C

170/

= x ln|x + V.x⁷ +11 - f    -

^{1 J}Vx² + i

= jt ln|jc -ł- -\/-v² + l|-^ J-j^fiL = xln|x + -Jx² + 1 -Vx‘’ +1 +C 171/

jjn|3 + 4.r[eZxr =

3 + 4x = / 4<Zv = dl

.    j

u = \nt du = — t

dv = dt v = /

= -/In/-— fdt =

i    i'

= -^(3 + 4x)ln|3 + 4x|--j-(3 + 4x)+C

f &

^J x(l + In^J

ln|x| = t *=dt

= f- ^ _T = arc/gt + C = arcfg (ln|x|)+ C ^J 1 + /

172/

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