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Basic Derivatives and Integrals
Basic Trigonometry Review
[sin x] dx = - cos x + C
d
dx [sin x] = cos x
[cos x] dx = sin x + C
d
2
dx [tan x] = sec x
[sec x] dx = tan x + C
d
2
dx [cot x] = csc x
[csc ] dx = cot x + C
d
dx [sec x] = sec x tan x
[sec x tan x] dx = sec x + C
d
dx [csc x] = csc x cot x
[csc x cot x] dx = csc x + C
d x
x
dx [e ] = e
[e ] dx = e + C
d
1
dx [ ln|x| ] = x
[ 1 ] dx = ln |x| + C
x
Adjacent
2
2
x
d
-1
1
dx [ tan x ] = 1 + x2
d
-1
1
dx [ sin x ] =
d
-1
dx [ sec x ] =
e
us
ten
po
Hy
1
sin x =
Opposite
Hypotenuse
1
cos x =
Adjacent
Hypotenuse
tan x =
Opposite
Adjacent
Basic Relationships
tan x =
sin x
cos x
cot x =
cos x
sin x
sec x =
1
cos x
csc x =
1
sin x
Trigonometric Identities
x
sin2x + cos2x = 1
tan2x + 1 = sec2x
1
-1
[
] dx = tan x + C
2
1+x
[
SOH-CAH-TOA
cot2x + 1 = csc2x
sin 2x = 2 sin x cos x
cos 2x = cos2x - sin2x
2
cos 2x = 2 cos x - 1
cos 2x = 1 - 2sin2x
sin 2x =
1
2
(1 - cos 2x)
cos2x =
1
2
(1 + cos 2x)
Addition Formulas
-1
] dx = sin x + C
sin( a + b) = sin a cos b + cos a sin b
sin( a - b) = sin a cos b - cos a sin b
1
[
-1
cos( a + b) = cos a cos b - sin a sin b
] dx = sec x + C
cos( a - b) = cos a cos b + sin a sin b
Mathematical Reference
Formulas and Constants
Quotient Rule
f(x)g(x)= f'(x)g(x) + f(x)g'(x)
f(x) f'(x)g(x) - f(x)g'(x)
=
g(x)
[ g(x) ]2
Chain Rule
Integration by Parts
uv - Ú v du
n-1
( f(x) ) = n( f(x) ) ( f'(x))
Electron mass: 9.109 x 10-31 kg
Gas Constant: 8.314 J(K-mole)-1
Electron Charge: 1.6 x 10-19 C
Avogrados Number: 6.022 x 1023
Proton mass: 1.672 x 10-27 kg
Grav Constant: 6.66 x 10-11 Nm2/Kg2
Permitivity Constant: 8.85 x 1012 C2/Nm2
ElectronVolt: 1.6 x 10-19 J
Speed of Light: 3 x 108 m/s
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More Integrals and Hyperbolics
Product Rule
n
Opposite
d
dx [cos x] = - sin x
Gravity: 9.8 m/s2
Speed of Sound: 344 m/s
SAP: 1.013 x 105 Pascals
[sin2x] dx = 12 x -
1
4
sin 2x + C
sinh x =
ex - e-x
2
cosh x =
ex + e-x
2
tanh x =
ex - e-x
ex + e-x
[sin3x] dx = 13 cos3x - cos x + C
[cos2x] dx = 12 x +
1
4
sin 2x + C
[cos3x] dx = sin x - 13 sin3x + C
[tan x] dx = ln | sec x | + C
[sec x] dx = ln | sec x + tan x | + C
[tan2x] dx = tan x - x + C
[sec2x] dx = tan x + C
[ln|x|] dx = x ln|x| - x + C