24 (3)

Biblioteczka Opracowań Matematycznych

PRZYKŁADY CAŁKOWANIA

149/    &x=/

[str4xdx=    —\tx=- \cH&xdx--dx=— \chtd    dl -~x =—sH&x—-x+C

^{J    J} 2    2^J    2    16^J dx=- 2    16    2

8

150/ jch² xdx- jchxc/fxdx - jc/u{l + shTx)jx = |5/lv = t chxdx = dt\ = J(l +f~ \łt =

= t + L + c =sfu + — + C 3    3

f—■ - = Lr = shl Vx² +1 = c/tf = chtd\ = f-= f-^- = -c/Af + C ³ x²\lx² +1    '    ' ^J shtchl ³ sh't

151/

+c=-^±L+c

$A/    X

chi

i52/_

fV X² — 1 I    r~5- ,    ,    . .1    1 rshtsht .    1 riA²/ , ,

|-£& = x = cAf Vx²-l = shr dx = sA/<# = — I—7—= — |——chtdi =

^J 5x I    i 5^J c-A/ 5 ³ch't

1 r sA²/

-U

5 ³ sh²t +1

-cAto7 =

shl = k chtdt = dk

= - f—^ -—arctgk+C = — sht--arcigsht+C -5^J*²+I    5    5    5    5

= 7\lx² -] - — arctgVx^: -1 +C 5    5

153/

r </r I    „ ,    .    , , , I r 2cA/rff 1 r dt

\-_r-    == x = 2shl Vx‘ + 4 = 2cA/ = 2cAfcA =■ j tj- = ~ I—777~

¹yj{x² + 4j ^{1    1}    (^2c/"?    ^{4 c}*^/

I .    _ 1 shl    1

= —tht + C =--+ C = —-_r

4    4 chi    4 /

154/

y+4

r + C = -

4 Vx²+ 4

+ C

j‘Vx⁷+9<& = |x = 3s/tf Vy+9=3 chi rft = 3cA/<*| = 9 JcA²/<* =9j

9 r    9 r    9    9    9    9    _ x-Jxⁱ + 9    9

= — [ chlldt + - f = -sA2/ + — 1 + C = -    + — / + C =---+ r In

2³    2^J    2    2    2    2 2    2

cA2/ +1

i-Vx^J + 9

+ C

9. Całki różne

155/= /,+/, + /,

/₂= j(x-X²)/v = -y+^- + C, /,= f(jT²-Ar)*r=y-y+C3

dla x e (-h,0] dla xe [0,1] dla x e [l,h)

Ostatecznie pozostało dobrać tak stałe Q , Cj, C_? aby funkcja podcałkowa pozostała ciągła w punktach x = 0 oraz x = 1.

Jeżeli C| = C = C2 oraz C3= l/3+C| to warunek ten będzie spełniony.

156/

f—^—-dx ³ x + -Jx + 2

x + 2 = f² = dx = 2 tdt x = r²- 2 t-1

= f f -' 2aft = 2 f /■— t/r = 2 f(r -1 + ^{21 2}    =

³1²-2+t    V+/-2 ^J l² +i-2

2i-2

_=t²-2t + 4\-r^—di=x+2-2^2+4\. \ ¹ >dt = x + 2-2&+2+4 f— = +r-2    ^J(r-lX/+2)    J/+2

=x+2-2Vx+2 + 4ln|/ + 2|+C = x+2-2>/x+2+4ln|Vx+2+2|+C 157/

f

x+ j2x-3 x-\

dx =

2x-3 = t² 2 dx = 2/<A t² + 3

r+3

r ->    a +2t² + 3t , _r 2/-2

=    —**= I r——<*= f(/ + 2 + ^—-)<*

³ r+3    ³ t +1 J r² + l

= £ + 2/ +    - 2    = ^£-2 + 272^3 + ln|/² +1| - 2arcigt+C =    + 271^3 +

2 V+l ^Jr+l 2    ¹¹    7

+ ln|2x - 2| - 2arctg -j2x-3 + C

158/ j-

dx

yjX² — X + \ = X-l

X = ■

X + 1

t² - 1

2/ -1

dx =

2(/²-/ + l) (2/-1)³

dt

-47-