Biblioteczka Opracowań Matematycznych

jVl -4.r²dx = j^4^-x¹'jdx = 2 jj^-x²dx =^arcsin 2x + x^~~x² +C

115/

116/

|VlO-+3.r-jr²tiT = jjA = *^_2~' = ljj-t²d' = yarcs.ry + i+ C

3

x —

= — arcsin ——- +--y]\0+3x-x² +C

8

117/

x + - = t 3

dx = dt

4}

U*

JV3T^;+iar+9A= J^x+|j +|<fr= fjl (.

= V3 J Jr*+Idr = V3 - .[t*+^+--tal

^JV 9 2V 9    2 9 V 9

hł

-TH—    +-

TH— +-dx =

("I)/

3x² + 10t + 9    V3,

-+—ln

3    9

3t² h-IOth-9

3T + 5 + -y/3(3T³ + 10T + 9)| + C

118/ r5T²-2-T+10    / _Ł\—i—o r

— - dx = (ax + bw3x -5th-8 + ^JV3t²-5t+8    ^J

Różniczkujemy obustronnie:

5t -2th- 10 V3t²-5t + 8

= ayl3x² -5t + 8 + (ot + />)-

+ C =

+ C =    3t³ + 10t + 9 +

6

Adx

-J3x² -5t+8

6t-5

2V3t²-5t + 8    V3t²-5t + 8

Obustronnie mnożymy przez ^3_X² - 5x + 8

5t² - 2t +10 = o(3t² - 5t + 8)+ (av +    + /t

5

a = —

6

12

/4 = — “ 24

Stąd układ równań:

10 = 12o

- 4 = -10a - 5a + 6A 20= I6a-5A + 2.4

Wstawiamy do wyjściowego wzoru:

, (5 17^1 tzi—:—" 55 _f dx

U 12)    8 ⁱyf^Z

i 55J3

5x + 8

i,-—I

¹    24 ^J

V 36

24

ln/ + ,|r+— 36

,0=^1,

24

6x-5

3x² - 5x + 8

5x + 8 ₊ ^Hl„3x-- + i/3y/3.x^!-5x + 8

24

119/ f * ^X+L<6: = (<rr² + 6x + c\lx¹ + 2x + 2 + A f-= ^JVx²+2x + 2

+ C civ

x³-x + l

+ 2x + 2

= (2av + 6)\/x² + 2x + 2 +(ar² +6x + c)—+

2>/x² +2x+2 l

ylx² + 2x+2

x³ -x + l = (2ax + 6)(x² + 2x + 2)+ (av² +bx + c)(x + l)+ A x³ -x +1 = x³(3a)+ x²(5a + 26)+ x(4a + 36 + c)+ (26 + c + A)

A

² +2x+2

3a = l 5<j + 26 = 0 4<j + 36 + c = -l 26 + c+. = 1

1

a = — 3

6 = — 6 1

c = — 6

6

/=flx²-lx + l-'l>/x² + 2x + 2 + — f-p=

U 6 ój    6^

dx

+ 2x+ 2

/,=lf-

¹ 0 J

dx

=u-

<ir

x +1 = / = <*

x² + 2x +2+ C

/ = ^x^J-lx + lW + 2x + 2 + l|n|x + l + >/x² + 2x + 2| + C