Common Derivatives and Integrals
Common Derivatives and Integrals
Derivatives
Integrals
Basic Properties/Formulas/Rules
d
( cf ( x ) ) = cf ¢ ( x ) , c is any constant. ( f ( x ) ± g ( x ) )¢ = f ¢ ( x ) ± g ¢ ( x )
dx
d n
d
( c ) = 0 , c is any constant.
( x ) = nx n-1 , n is any number.
dx
dx
æ f ö¢ f ¢ g - f g ¢
– (Quotient Rule)
( f g )¢ = f ¢ g + f g ¢ – (Product Rule) ç ÷ =
g2
ègø
d
f ( g ( x ) ) = f ¢ ( g ( x ) ) g ¢ ( x ) (Chain Rule)
dx
g¢ ( x)
d
d g (x)
g x
ln g ( x ) ) =
e
= g¢ ( x) e ( )
(
dx
g ( x)
dx
(
)
( )
Common Derivatives
Polynomials
d
d
(c) = 0
( x) = 1
dx
dx
d
( cx ) = c
dx
Trig Functions
d
( sin x ) = cos x
dx
d
( sec x ) = sec x tan x
dx
d
( cos x ) = - sin x
dx
d
( csc x ) = - csc x cot x
dx
d
( tan x ) = sec 2 x
dx
d
( cot x ) = - csc 2 x
dx
Inverse Trig Functions
d
1
sin -1 x ) =
(
dx
1 - x2
d
( sec -1 x ) = 12
dx
x x -1
d
1
cos -1 x ) = (
dx
1 - x2
d
( csc-1 x ) = - 12
dx
x x -1
d
1
tan -1 x ) =
(
dx
1 + x2
d
1
( cot -1 x ) = - 1 + x 2
dx
d n
( x ) = nx n-1
dx
d
( cx n ) = ncx n -1
dx
Basic Properties/Formulas/Rules
ò cf ( x ) dx = c ò f ( x ) dx , c is a constant.
b
b
ò a f ( x ) dx = F ( x ) a = F ( b) - F ( a ) where
b
ò f ( x ) ± g ( x ) dx = ò f ( x ) dx ± ò g ( x ) dx
F ( x ) = ò f ( x ) dx
b
b
b
b
ò a cf ( x ) dx = c ò a f ( x ) dx , c is a constant. ò a f ( x ) ± g ( x ) dx = ò a f ( x ) dx ± ò a g ( x ) dx
a
b
ò a f ( x ) dx = 0
b
a
ò a f ( x ) dx = -ò b f ( x ) dx
c
b
b
ò a f ( x ) dx = ò a f ( x ) dx + ò c f ( x ) dx
If f ( x ) ³ 0 on a £ x £ b then
ò a c dx = c ( b - a )
b
ò a f ( x ) dx ³ 0
If f ( x ) ³ g ( x ) on a £ x £ b then
b
b
ò a f ( x ) dx ³ ò a g ( x ) dx
Common Integrals
Polynomials
1
ò dx = x + c
ò k dx = k x + c
ò x dx = n + 1 x
ó 1 dx = ln x + c
ô
õx
òx
òx
-1
n
dx = ln x + c
ó 1 dx = 1 ln ax + b + c
ô
õ ax + b
a
òx
p
q
-n
dx =
p
q
+ c, n ¹ -1
1
x - n +1 + c, n ¹ 1
-n + 1
p
dx =
n +1
+1
q
1
xq +c =
x
+1
p+q
Trig Functions
ò cos u du = sin u + c
p+ q
q
+c
ò sin u du = - cos u + c
ò sec u du = tan u + c
ò sec u tan u du = sec u + c ò csc u cot udu = - csc u + c ò csc u du = - cot u + c
ò tan u du = ln sec u + c
ò cot u du = ln sin u + c
1
ò sec u du = ln sec u + tan u + c
ò sec u du = 2 ( sec u tan u + ln sec u + tan u ) + c
2
2
3
Exponential/Logarithm Functions
d x
d x
( a ) = a x ln ( a )
(e ) = ex
dx
dx
d
1
d
1
( ln ( x ) ) = x , x > 0
( ln x ) = x , x ¹ 0
dx
dx
Hyperbolic Trig Functions
d
( sinh x ) = cosh x
dx
d
( sech x ) = - sech x tanh x
dx
ò csc u du = ln csc u - cot u + c
d
1
( log a ( x ) ) = x ln a , x > 0
dx
Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes.
d
( tanh x ) = sech 2 x
dx
d
( coth x ) = - csch 2 x
dx
© 2005 Paul Dawkins
3
u du =
1
( - csc u cot u + ln csc u - cot u ) + c
2
Exponential/Logarithm Functions
u
u
ò e du = e + c
d
( cosh x ) = sinh x
dx
d
( csch x ) = - csch x coth x
dx
ò csc
u
ò a du =
au
+c
ln a
e au
( a sin ( bu ) - b cos ( bu ) ) + c
a + b2
e au
au
e
cos
bu
du
=
(
)
( a cos ( bu ) + b sin ( bu ) ) + c
ò
a 2 + b2
òe
au
sin ( bu ) du =
2
Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes.
ò ln u du = u ln ( u ) - u + c
ò ue du = ( u - 1) e
u
u
+c
ó 1 du = ln ln u + c
ô
õ u ln u
© 2005 Paul Dawkins
Common Derivatives and Integrals
Inverse Trig Functions
1
ó
æuö
du = sin -1 ç ÷ + c
ô
2
2
èaø
õ a -u
1
ó 1
æuö
du = tan -1 ç ÷ + c
ô 2
2
a
õ a +u
èaø
1
1
ó
æuö
du = sec -1 ç ÷ + c
ô
a
èaø
õ u u2 - a2
Hyperbolic Trig Functions
ò sinh u du = cosh u + c
ò cosh u du = sinh u + c
ò tanh u du = ln ( cosh u ) + c
ò sin
u du = u sin -1 u + 1 - u 2 + c
-1
ò tan
1
u du = u tan -1 u - ln (1 + u 2 ) + c
2
-1
ò cos
u du = u cos -1 u - 1 - u 2 + c
-1
ò sech u tanh u du = - sech u + c
ò csch u coth u du = - csch u + c
ò sech u du = tan sinh u + c
Miscellaneous
ó 1 du = 1 ln u + a + c
ô 2
õ a - u2
2a u - a
ò
2au - u 2 du =
ò
2
u du = tanh u + c
2
u du = - coth u + c
ó 1 du = 1 ln u - a + c
ô 2
õ u - a2
2a u + a
ò
ò
ò sech
ò csch
-1
u 2
a2
a + u du =
a + u 2 + ln u + a 2 + u 2 + c
2
2
2
u
a
u 2 - a 2 du =
u 2 - a 2 - ln u + u 2 - a 2 + c
2
2
u 2
a2
æuö
2
2
2
a - u du =
a - u + sin - 1 ç ÷ + c
2
2
èaø
2
Common Derivatives and Integrals
fraction decomposition of the rational expression. Integrate the partial fraction
decomposition (P.F.D.). For each factor in the denominator we get term(s) in the
decomposition according to the following table.
Factor in Q ( x )
2
u Substitution
ò a f ( g ( x ) ) g ¢ ( x ) dx then the substitution u = g ( x ) will convert this into the
b
g(b)
integral, ò f ( g ( x ) ) g ¢ ( x ) dx = ò
f ( u ) du .
a
g (a)
b
Integration by Parts
The standard formulas for integration by parts are,
ò udv = uv - ò vdu
b
b
b
ò a udv = uv a - ò a vdu
Choose u and dv and then compute du by differentiating u and compute v by using the
fact that v = ò dv .
Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes.
Term in P.F.D Factor in Q ( x )
ax + b
A
ax + b
ax 2 + bx + c
Ax + B
ax + bx + c
2
( ax + b )
( ax
2
Term in P.F.D
A1
A2
Ak
+
+L +
k
ax + b ( ax + b )2
( ax + b )
k
+ bx + c )
Ak x + Bk
A1 x + B1
+L +
k
2
ax + bx + c
( ax 2 + bx + c )
k
Products and (some) Quotients of Trig Functions
n
m
ò sin x cos x dx
u-a
a2
æ a -u ö
2au - u 2 + cos -1 ç
÷+c
2
2
è a ø
Standard Integration Techniques
Note that all but the first one of these tend to be taught in a Calculus II class.
Given
Trig Substitutions
If the integral contains the following root use the given substitution and formula.
a
a2 - b2 x2
Þ
x = sin q
and
cos 2 q = 1 - sin 2 q
b
a
b2 x2 - a 2
x = sec q
and
tan 2 q = sec 2 q - 1
Þ
b
a
Þ
a 2 + b2 x 2
x = tan q
and
sec 2 q = 1 + tan 2 q
b
Partial Fractions
ó P ( x)
If integrating ô
dx where the degree (largest exponent) of P ( x ) is smaller than the
õ Q ( x)
degree of Q ( x ) then factor the denominator as completely as possible and find the partial
© 2005 Paul Dawkins
1. If n is odd. Strip one sine out and convert the remaining sines to cosines using
sin 2 x = 1 - cos 2 x , then use the substitution u = cos x
2. If m is odd. Strip one cosine out and convert the remaining cosines to sines
using cos 2 x = 1 - sin 2 x , then use the substitution u = sin x
3. If n and m are both odd. Use either 1. or 2.
4. If n and m are both even. Use double angle formula for sine and/or half angle
formulas to reduce the integral into a form that can be integrated.
n
tan
x
sec m x dx
ò
1. If n is odd. Strip one tangent and one secant out and convert the remaining
tangents to secants using tan 2 x = sec 2 x - 1 , then use the substitution u = sec x
2. If m is even. Strip two secants out and convert the remaining secants to tangents
using sec 2 x = 1 + tan 2 x , then use the substitution u = tan x
3. If n is odd and m is even. Use either 1. or 2.
4. If n is even and m is odd. Each integral will be dealt with differently.
Convert Example : cos 6 x = ( cos 2 x ) = (1 - sin 2 x )
3
3
Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes.
© 2005 Paul Dawkins