wzory_pochodne_i_calki.pdf

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78193785 UNPDF
Wzorynapochodneicałki
( c ) 0 =0 ,c –stałarzeczywista
Z
x dx = x +1
( x ) 0 = x − 1 , 2 R
+1 + C 6 = − 1
( a x ) 0 = a x ln a,a> 0 ,a 6 =1
Z
( e x ) 0 = e x
e x dx = e x + C
Z
(sin x ) 0 =cos x
cos xdx =sin x + C
Z
(cos x ) 0 = − sin x
sin xdx = − cos x + C
(tg x ) 0 = 1
cos 2 x
Z 1
sin 2 x dx = − ctg x + C
(ctg x ) 0 = − 1
sin 2 x
(log a x ) 0 = 1
x ln a ,a> 0 ,a 6 =1
(ln x ) 0 = 1
x ,x> 0
Z 1
x dx =ln | x | + C
Z 1
(arcsin x ) 0 = 1
p 1 − x 2 , − 1 <x< 1
p 1 − x 2 dx =arcsin x + C
(arccos x ) 0 = − 1
p
1 − x 2 , − 1 <x< 1
Z 1
x 2 +1 dx =arctg x + C
(arctg x ) 0 = 1
x 2 +1
(arcctg x ) 0 = − 1
x 2 +1
(sh x ) 0 =ch x
Z
ch xdx =sh x + C
Z
(ch x ) 0 =sh x
sh xdx =ch x + C
Z dx
p x 2 + K =ln | x +
p
x 2 + K | + C,K 2 R
Z p
p
p
x 2 + Kdx = K
x 2 + K | + x
2
2 ln | x +
x 2 + K + C,K 2 R
Z dx
p
a 2 − x 2 =arcsin x
a + C,a> 0
Z p
a 2 − x 2 dx = a 2
2 arcsin x
a + x
p
a 2 − x 2 + C,a> 0
2
Z 1
cos 2 x dx =tg x + C
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