13 (10)

Biblioteczka Opracowań Matematycznych

85/ r__;

Ux-

x-4

x-4

(*-2X*-3)

A

~dx — /

B _ x(A + B)-3A~2B

(x-2)(x-3) x — 2    x-3

/4 = 2

(x-2Xx-3)

.4 + 5 = 1 -3/1-25 =-4 dx

B = -1

^{/ =} 2^"^⁼'^{21n|jC_2|_ln|j;}"^{3| +} C

86/

r 2x + 7 ,    r ZX +1 + O , r ZX+ l , r

I—-dr = I—-<&= i—-dx +

•*x"+x —2    3r' + r —2    ■'x² + x-2 J

¹ x +x-2

r 2x + 1 + 6

—=-1

’x² + x-2

2x + l x² +x-2

6dx

¹ x² + x- 2

'= J—T + J—-.-2ln|, ₊ 2|₊2ln|»-!|₊c

J

= In|x² +x-2| + /

6    /I 5    x(/l + 5)-/l + 25

x^:+x-2 x + 2 x — 1    (x + 2X*~l)

A + B = 0    JA=—2

-A + 2B = 6    [5 = 2

r- 2dx r 2<&

x + 2 ^Jx-1 Y~⁺ ^    = ln|x² + x-2|-2In|x + 2|+21n|x-l| + C

87/

J

3x² + 2x-3

x³-x

3x -1 + 2x- 2 x(x-l)

:    r3x‘ -1 r 2x

J——<*+hr

^J x -x ^J xlx

2x-2

V⁷-*/

dx

= ln|x³ -x| + 2 J-

x(x +1)

ln x - x + /

_1_₌A₊_B_ ₌ x(A + B)+ A

x(x + l) X x + l    x(x + l)

\A+B = 0    j A = \

[ A = 1    [5 = -l

^{/ = 2}jf"²^ ^{= 2taH_2,n|x+l1}

f

3x² + 2x - 3

dx- ln|x³ -x|+2ln|x|-21n|x + l|+C

m J^^⁼(^x+4+5^}^&=y^+4x+I^r7^dr=T^+4x+/

7x +1 7x + l /< _|    5 _x(,4+5)-5

(

x(x-l) X x-l x(x-l) A + B = 7    { A = &

-5=1    |5 = -|

^/=J~“f7rr^=8lnN-^lnk-^,l^+c

J%^^ = ^- + 4x-8ln|x|-ln|x-l| + C

m h

dr

= /

\'^{X X}A ^{+ X}B Cx + D x\A + C)-x'(A-B-D)+x{A-B)+B

\    i i    .    77 ł    n    "

x³(x³-x + l) x x³ x³-x+l

³(x³ -x + l)

A + C = O -/4 + 5+D = 0 A-B = 0 5 = 1

t£c r<Ar

-4 = 1 5 = 1

C = -l 0 = 0 xdx

l = f« + J    - = ln|x|--—/,

^J X ^Jx² Jx³-x + l    ¹¹ X

.<-3

1 1

J f ₌ f 2 ⁺ 3_A=l ęyji4f_T^--^-xłi4f

J r² — r+1 •'r² —r + 1    2^x³ — X + 1 2 ^ —X+1 2    2^1

dx

J.t²-x+l    x* —x+l    2-*x³-x+1 2-’x³-x+1 2

= | ln|x³ - x + J| +    +C

/ = ln|x| - ^ ^ ln|x³ - x +1| - -y arctg    + C

H)

x-2 = / dx-dt x=t+2

90/ o

f ^{2x 5} dx = \dx = dt

^J(^2p

₌ J^±^₌|2^9_{a = 2}J^₊₉J^ _{= 21lł})_?_+c

-25-