Derivative and Integral Formulas
Calculus I-IV ~ Prof. Sally J. Keely, M.S.
DERIVATIVE
FUNCTION
INTEGRAL
ex
ex
ex
x
b ln b
1
x
1
x ln b
b
bx
ln b
x
ln x
x(ln x 1)
logb x
x
(ln x 1)
ln b
cos x
sin x
cos x
sin x
cos x
sin x
sec2 x
tan x
ln sec x
csc2 x
sec x tan x
csc x cot x
sin 2x
sin 2x
2 tan x sec2 x
2cot x csc2 x
2 tan x sec2 x
2cot x csc2 x
cot x
ln csc x
ln sin x
sec x
ln sec x tan x
csc x
ln csc x cot x
sin 2 x
1
2
x
1
4
sin 2 x
cos2 x
1
2
x
1
4
sin 2 x
tan 2 x
cot 2 x
sec2 x
tan x x
cot x x
tan x
csc2 x
cot x
cosh x
sinh x
cosh x
sinh x
cosh x
sinh x
sech 2 x
tanh x
ln(cosh x )
csch 2 x
coth x
ln sinh x
sech x tanh x
sech x
arctan(sinh x )
csch x coth x
csch x
ln tanh 12 x
Copyright
1992 Sally J. Glover
All Rights Reserved.
151-forms_derivinteg.docx
revised 2010-01-30
Derivative and Integral Formulas (Calc I version)
Calculus I-IV ~ Prof. Sally J. Keely, M.S.
DERIVATIVE
FUNCTION
1
1 x
2
1
1 x2
1
x
2
x
2
1
1
1
1
1
1
( x 2 1)
2
1 x
1
( x 2 1)
2
1 x
1
arcsinh x
x arcsinh x
1
x
x
2
**
1
1
** Note:
x x
2
dx
x 2 1 (arccosh x
0)
x arccosh x
x 2 1 ( arccosh x
0)
x arctanh x
arccoth x
x arccoth x
arccsch x
x2 1
x arccosh x
arctanh x
arcsech x
x 1 x2
1
ln( x 2 1)
2
1
x arccot x
ln( x 2 1)
2
x arctan x
x arccsc x arccosh x
arccosh x
1
1 x2
arccsc x
x x2 1
x
x arccos x
x arcsec x arccosh x
1
2
arccos x
1 x2
arcsec x
x x2 1
x
x arcsin x
arccot x
1
2
arcsin x
arctan x
1
INTEGRAL
1
ln(1 x 2 )
2
1
ln( x 2 1)
2
x arcsech x arcsin x
x arccsch x
x
arcsinh x
x
arccsch x
1
Copyright
1992 Sally J. Glover
All Rights Reserved.
151-forms_derivinteg.docx
revised 2010-01-30
Derivative and Integral Formulas (Calc II version)
Calculus I-IV ~ Prof. Sally J. Keely, M.S.
DERIVATIVE
FUNCTION
1
a
2
x
a
x arcsin
arccos
x
a
x arccos
arctan
x
a
x arctan
x
a
a
ln( x 2 a 2 )
2
arccot
x
a
x arccot
x
a
a
ln( x 2
2
a2 )
a
2
arcsec
x
a
x arcsec
x
a
a arccosh
x
a
a
2
arccsc
x
a
x arccsc
x
a
a arccosh
x
a
1
a
2
x
a
x
2
a
2
a
x a2
2
a
x x
2
x x
2
a
1
x
2
a
arcsinh
2
1
a
2
x
a
x2
a2
x2
x arcsinh
x
a
x2 a2
x arccosh
x
a
x2
a 2 (arccosh
x
a
0)
x arccosh
x
a
x2
a 2 ( arccosh
x
a
0)
x
( x2
a2 )
arctanh
x
a
x arctanh
x
a
a
ln(a 2
2
x
2
( x2
a2 )
arccoth
x
a
x arccoth
x
a
a
ln( x 2 a 2 )
2
arcsech
x
a
x arcsech
x
a
a arcsin
arccsch
x
a
a
a
x a
2
x
a
x
a2
2
a
a
x
a
x
arccosh
a
x2 a2
2
x
a
arcsin
2
x
2
INTEGRAL
x
2
a
2
2
**
a
** Note:
x x
2
a
2
dx
arccsch
Copyright
1992 Sally J. Glover
All Rights Reserved.
x arccsch
x
a
x2 )
x
a
ax
x
arcsinh
x
a
x
a
151-forms_derivinteg.docx
revised 2010-01-30