Diff and Integ Rules Final

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Transcription

Diff and Integ Rules Final
Differentiation Formula
(chain rule form)
Integration Formula for f (u)
d
 ku  ku'
k = constant
dx
d
kf (u)  kf '(u)
dx
d
 f (u)  g(u)  f '(u)  g'(u)
dx
 du  u  C
d
 u  v   u  v '  v  u'
dx
d  u  v  u'  u  v '

dx  v 
v2
 u dv  u  v   v du
d n
u   nu n 1u'

dx
d
 sin u  u' cos u
dx
d
cos u  u' sin u
dx
d
 tan u  u' sec 2 u
dx
d
 sec u  u' sec u tan u
dx
d
 cot u  u' csc 2 u
dx
d
 csc u  u' csc u cot u
dx
(First) Fundamental Theorem of Calculus:
b
 f ( x) dx  F (b)  F (a)
a
Second Fundamental Theorem of Calculus:
u = g(x)
 kf (u) du  k  f (u) du  C
  f (u)  g (u) du   f (u) du   g (u) du
 u n du 
u n 1
C
n 1
(integration by parts)
n  1
 cos u du  sin u  C
 sin u du   cos u  C
 sec
2
u du  tan u  C
 sec u tan u du  sec u  C
 csc
2
u du   cot u  C
 csc u cot u du   csc u  C
 tan u du  ln sec u  C   ln cos u  C
 cot u du   ln csc u  C  ln sin u  C
 sec u du  ln sec u  tan u  C   ln sec u  tan u  C
 csc u du   ln csc u  cot u  C  ln csc u  cot u  C
u
 f (t ) dt  u ' f (u )  v' f (v)
v
where u  g (x) and v  h(x)
2
 arcsin u du  u arcsin u  1  u  C
 arccos u du  u arccos u  1  u  C
 arctan u du  u arctan u  ln u  1  C
 arc cot du  u arccot u  ln u  1  C
 arcsec du  u arcsec u  ln u  u  1  C
2
2
2
2
 arccsc u du  u arccsc u  ln
u  u2  1  C
d
u'
arcsin u 

dx
1  u2
d
u'
arctan u 

dx
1  u2
d
u'
arcsec u 

dx
u u2  1









d

 arccos u

dx

d

 arccot u

dx

d

 arccsc u

dx

u

2
2
dx  
du
2
2
u
u
u 2  a2
dx  
du
u u 2  a2
au
 C or
ln a
u
u
 e du  e  C
d
u'
 log a u  
dx
u ln a
 ln u du  u ln u  u  C
 
u
C
a
a u
a u

u
du
1
u
 a 2  u 2 dx   a 2  u 2  a arctan a  C
d u
a  u'a u ln a
dx
d u
e  u'e u
dx
d
u'
ln u 

dx
u
 
 arcsin
u
 a du 
u
1
 ua
 u dx  u du  ln u  C
u

u
1
arcsec  C
a
a
ln a dx  a u  C

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