CALKA N I E OZN ACZON A
D E FIN ICJA .
N ie c h f b �e d z ie fu n kc ja� o kr e �s lo n a� w p e wn ym p r z e d z ia le I. Calk� nieoznaczon�
a a
fu n kc ji f n a z ywa m y ka zŁd a� fu n kc j�e F r �o zŁn ic z ko wa ln a� w I i s p e ln ia ja� c a� d la ka zŁd e g o
x " I wa r u n e k F2 ( x) = f( x) .
P is z e m y: F( x) = f( x) dx. M�o wim y, zŁe f je s t c a lko wa ln a w I.
U W A GA 1 . Gd y F( x) je s t c a lk� fu n kc ji f( x) , t o F( x) + C t e zŁ je s t c a lk� t e j
a a
fu n kc ji.
W yjaśnienie: [F( x) + C]2 = F2 ( x) + C2 = F2 ( x) + 0 = F2 ( x) = f( x) .
U W A GA 2 . D wie c a lki F1( x) i F2( x) t e j s a m e j fu n kc ji f( x) r �o zŁn i� s i�e w c a lym
a
p r z e d z ia le I o p e wn a� s t a la� .
W yjaśnienie: F12 ( x) = f( x) o r a z F22 ( x) = f( x) , wi�e c [F1( x) - F2( x) ]2 = 0 ; z
wn io s ku 3 p o t wie r d z e n iu L a g r a n g e  a wie m y, zŁe je d yn ie p o c h o d n a fu n kc ji s t a le j
je s t s t a le r �o wn a z e r o , c z yli is t n ie je t a ka s t a la C, zŁe F1( x) - F2( x) = C.
W N IOS E K . Zn a ja� c je d n a� c a lk�e F fu n kc ji f o t r z ym a m y ws z ys t kie p o z o s t a le :
f( x) dx = F( x) + C.
TW IE R D ZE N IE . K a zŁd a fu n kc ja c i� g la je s t c a lko wa ln a .
a
P OD S TA W OW E W ZOR Y ( c z �e �s �c p ie r ws z a ) .
Ca lko wa n ie o d n o s i s i�e d o t yc h p r z e d z ia l�o w, w kt �o r yc h fu n kc je p o d c a lko we s a�
o kr e �s lo n e .
1
xadx = xa+1 + C d la a = -1 ( 1 )

a + 1
1
dx = ln |x| + C ( 2 )
x
exdx = ex + C ( 3 )
ax
axdx = + C ( 4 )
ln a
s in xdx = - c o s x + C ( 5 )
c o s xdx = s in x + C ( 6 )
1
dx = t g x + C ( 7 )
2
c o s x
1
dx = -c t g x + C ( 8 )
2
s in x
1
dx = a r c t g x + C = -a r c c t g x + K ( 9 )
1 + x2
1
1
" dx = a r c s in x + C = - a r c c o s x + K ( 1 0 )
1 - x2
W L A S N OS� CI.
[f( x) ą g( x) ]dx = f( x) dx ą g( x) dx
f( x) dx =  f( x) dx
f2 ( x) dx = f( x) + C
2
f( x) dx = f( x)
CA L K OW A N IE P R ZE Z CZE� S� CI:
u( x) v2 ( x) dx = u( x) v( x) - u2 ( x) v( x) dx
Za kla d a m y t u , zŁe fu n kc je u( x) i v( x) m a j� c ia� g le p o c h o d n e .
a
W yprowadzenie wzoru:
2
( uv) = u2 v + uv2
2
( uv) dx = ( u2 v + uv2 ) dx
uv = u2 vdx + uv2 dx
uv2 dx = uv - u2 vdx
P R ZY K L A D .
u=x v2 =ex
xexdx = = xex - 1 � exdx = xex - ex + C
u2 =1 v=ex
CA L K OW A N IE P R ZE Z P OD S TA W IE N IE :
f( x) dx = f[Ć( t) ]Ć2 ( t) dt, g d z ie x = Ć( t)
Za kla d a m y t u , zŁe fu n kc ja Ć : ( ą, �) ( a,b) m a c ia� g la� p o c h o d n a� Ć2 o r a z zŁe
fu n kc ja f : ( a, b) R je s t c ia� g la .
W yprowadzenie wzoru: N ie c h F( x) = f( x) dx ( o z n a c z a t o , zŁe F2 ( x) = f( x) ) .
Fu n kc ja z lo zŁo n a F[Ć( t) ] m a p o c h o d n a� ( wz g l�e d e m t)
2
F[Ć( t) ] = F2 [Ć( t) ]Ć2 ( t) = f[Ć( t) ]Ć2 ( t) .
Za t e m ,
f[Ć( t) ]Ć2 ( t) dt = F[Ć( t) ] = F( x) = f( x) dx.
P R ZY K L A D .
1 1 1
x=3t
c o s xdx = = ( c o s � 3 t) 3 dt = 3 c o s tdt = 3 s in t+C = 3 s in x+C
dx=3dt
3 3 3
 1
x=3t x=t
3
Zwykle z a m ia s t p is z e m y .
1
dx=3dt dx=dt dx=3dt
3
P R ZY K L A D .
1
1 1
x+7=t
5
s in x+7 dx = = ( s in t) 5 dt = -5 c o s t+C = -5 c o s x+7 +C
1
dx=dt dx=5dt
5 5 5
P OD S TA W OW E W ZOR Y ( c z �e �s �c d r u g a ) .
Ca lko wa n ie o d n o s i s i�e d o t yc h p r z e d z ia l�o w, w kt �o r yc h fu n kc je p o d c a lko we s a�
o kr e �s lo n e .
g2 ( x)
dx = ln |g( x) | + C ( 1 1 )
g( x)
ln xdx = x ln x - x + C ( 1 2 )
t g xdx = - ln | c o s x| + C ( 1 3 )
c t g xdx = ln | s in x| + C ( 1 4 )
"
a r c s in xdx = x a r c s in x + 1 - x2 + C ( 1 5 )
1
a r c t g xdx = xa r c t g x - ln ( 1 + x2) + C ( 1 6 )
2
dx x
= a r c s in + C ( 1 7 )
"
q
q - x2
dx
= ln |x + q + x2| + C ( 1 8 )
q + x2
1 1 x
q - x2dx = x q - x2 + q a r c s in + C ( 1 9 )
"
2 2 q
1 1
q + x2dx = x q + x2 + q ln |x + q + x2| + C ( 2 0 )
2 2
1 1
2
s in xdx = x - s in 2 x + C ( 2 1 )
2 4
1 1
2
c o s xdx = x + s in 2 x + C ( 2 2 )
2 4
W ZOR Y R E K U R E N CY JN E :
dx n n
Ca lki In = , Jn = s in xdx, Kn = c o s xdx p o t r a fi m y o b lic z y�c d la
(1+x2)n
n = 1 o r a z d la n = 0 . D la n e" 2 s t o s u je m y wz o r y:
1 x 2 n - 3
In = � + In-1
n-1
2 n - 2 ( 1 + x2) 2 n - 2
1 n - 1
n-1
Jn = - c o s x s in x + Jn-2
n n
1 n - 1
n-1
Kn = s in x c o s x + Kn-2
n n


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