Biblioteczka Opracowań Matematycznych
jVl -4.r2dx = j^4^-x1'jdx = 2 jj^-x2dx =^arcsin 2x + x^~~x2 +C
3
x —
= — arcsin ——- +--y]\0+3x-x2 +C
8
117/
x + - = t 3
-TH— +-
TH— +-dx =
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 ^3X2 - 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
V 36
24
ln/ + ,|r+— 36
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
dx
<ir
x +1 = / = <*
x2 + 2x +2+ C
/ = ^xJ-lx + lW + 2x + 2 + l|n|x + l + >/x2 + 2x + 2| + C
-37-