This document provides summaries of common derivatives and integrals, including: - Basic properties and formulas for derivatives and integrals of functions like polynomials, trig functions, inverse trig functions, exponentials/logarithms, and more. - Standard integration techniques like u-substitution, integration by parts, and trig substitutions. - How to evaluate integrals of products and quotients of trig functions using properties like angle addition formulas and half-angle identities. - How to use partial fractions to decompose rational functions for the purpose of integration. So in summary, this document outlines essential derivatives and integrals for many common functions, along with standard integration strategies and techniques.

1. Common Derivatives and Integrals Common Derivatives and Integrals
Derivatives Integrals
Basic Properties/Formulas/Rules Basic Properties/Formulas/Rules
d
( cf ( x ) ) = cf ¢ ( x ) , c is any constant. ( f ( x ) ± g ( x ) )¢ = f ¢ ( x ) ± g ¢ ( x ) ò cf ( x ) dx = c ò f ( x ) dx , c is a constant. ò f ( x ) ± g ( x ) dx = ò f ( x ) dx ± ò g ( x ) dx
dx b b
d n
( x ) = nxn-1 , n is any number. d
( c ) = 0 , c is any constant. ò a f ( x ) dx = F ( x ) a = F (b ) - F ( a ) where F ( x ) = ò f ( x ) dx
dx dx b 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 ± òa g ( x ) dx
æ f ö¢ f ¢ g - f g ¢
( f g )¢ = f ¢ g + f g ¢ – (Product Rule) ç ÷ = – (Quotient Rule) a b a
ègø g2 ò a f ( x ) dx = 0 ò a f ( x ) dx = -òb f ( x ) dx
d
( )
f ( g ( x ) ) = f ¢ ( g ( x ) ) g ¢ ( x ) (Chain Rule)
b c b b
dx ò a f ( x ) dx = ò a f ( x ) dx + òc f ( x ) dx ò a c dx = c ( b - a )
( ln g ( x )) = g (x )
g¢ x b
dx
e( )
d g ( x)
= g ¢( x ) e ( )
g x d
dx ( )
If f ( x ) ³ 0 on a £ x £ b then ò a f ( x ) dx ³ 0
b b
If f ( x ) ³ g ( x ) on a £ x £ b then ò a f ( x ) dx ³ ò a g ( x ) dx
Common Derivatives
Polynomials Common Integrals
d d d d n d
dx
(c) = 0
dx
( x) = 1
dx
( cx ) = c
dx
( x ) = nxn-1 dx
( cxn ) = ncxn -1 Polynomials
1
ò dx = x + c ò k dx = k x + c ò x dx = n + 1 x
n+1
n
+ c, n ¹ -1
Trig Functions
ó 1 dx = ln x + c 1
òx òx
-n
d d d ô
-1
dx = ln x + c dx = x - n +1 + c , n ¹ 1
( sin x ) = cos x ( cos x ) = - sin x ( tan x ) = sec2 x õx -n + 1
dx dx dx p p p+q
ó 1 dx = 1 ln ax + b + c 1 q +1 q
d d d
òx dx = x +c= +c
q q
( sec x ) = sec x tan x ( csc x ) = - csc x cot x ( cot x ) = - csc2 x ô x
dx dx dx õ ax + b a p
q +1 p+ q
Inverse Trig Functions Trig Functions
d
( sin -1 x ) = 1 2 d
( cos-1 x ) = - 1 2 d
( tan -1 x ) = 1 +1x2 ò cos u du = sin u + c ò sin u du = - cos u + c ò sec u du = tan u + c
2
dx 1- x dx 1- x dx
ò sec u tan u du = sec u + c ò csc u cot udu = - csc u + c ò csc u du = - cot u + c
2
d
dx
( sec-1 x ) = 12 d
dx
( csc-1 x ) = - 12 d
dx
( cot -1 x ) = - 1 +1x2 ò tan u du = ln sec u + c ò cot u du = ln sin u + c
x x -1 x x -1
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
3
Exponential/Logarithm Functions
d x d x 1
( a ) = a x ln ( a ) (e ) = ex ò csc u du = ln csc u - cot u + c ò csc
3
u du =
2
( - csc u cot u + ln csc u - cot u ) + c
dx dx
d
( ln ( x ) ) = 1 , x > 0 d
( ln x ) = 1 , x ¹ 0
d 1
dx x dx x dx
( log a ( x ) ) = x ln a , x > 0 Exponential/Logarithm Functions
au
ò e du = e + c ò a du = +c ò ln u du = u ln ( u ) - u + c
u u u
Hyperbolic Trig Functions ln a
d d d e au
( sinh x ) = cosh x ( cosh x ) = sinh x ( tanh x ) = sech 2 x òe
au
sin ( bu ) du = ( a sin ( bu ) - b cos ( bu ) ) + c ò ue du = ( u - 1) e
u u
+c
dx dx dx a + b2 2
d d d
( sech x ) = - sech x tanh x ( csch x ) = - csch x coth x ( coth x ) = - csch 2 x eau cos ( bu ) du = 2
e au
( a cos ( bu ) + b sin ( bu ) ) + c ó 1 du = ln ln u + c
dx dx dx ò a + b2
ô
õ u ln u
Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins

2. Common Derivatives and Integrals Common Derivatives and Integrals
Inverse Trig Functions
ó 1 æu ö Trig Substitutions
du = sin -1 ç ÷ + c ò sin
-1
ô u du = u sin -1 u + 1 - u 2 + c If the integral contains the following root use the given substitution and formula.
õ a -u2 2
èaø
a
ó 1 1 æuö 1 a 2 - b2 x2 Þ x = sin q and cos2 q = 1 - sin 2 q
ô 2 du = tan -1 ç ÷ + c ò tan
-1
u du = u tan -1 u - ln (1 + u 2 ) + c b
õ a +u
2
a èaø 2 a
b2x2 - a 2 Þ x = sec q and tan 2 q = sec 2 q -1
ó 1 1 æuö b
du = sec-1 ç ÷ + c ò cos
-1
ô u du = u cos - 1 u - 1 - u 2 + c
õ u u2 - a2 a èaø a
a2 + b2 x2 Þ x = tan q and sec 2 q = 1 + tan 2 q
b
Hyperbolic Trig Functions Partial Fractions
ò sinh u du = cosh u + c ò cosh u du = sinh u + c ò sech u du = tanh u + c ó P ( x)
2
If integrating ô dx where the degree (largest exponent) of P ( x ) is smaller than the
ò sech tanh u du = - sech u + c ò csch coth u du = - csch u + c ò csch
2
u du = - coth u + c õ Q (x)
degree of Q ( x ) then factor the denominator as completely as possible and find the partial
ò tanh u du = ln ( cosh u ) + c ò sech u du = tan sinh u + c
-1
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
Miscellaneous
decomposition according to the following table.
ó 1 du = 1 ln u + a + c ó 1 du = 1 ln u - a + c
ô 2 ô 2
õ a - u2 2a u - a õ u - a2 2a u + a Factor in Q ( x ) Term in P.F.D Factor in Q ( x ) Term in P.F.D
u a2
ò a + u du = a 2 + u 2 + ln u + a 2 + u 2 + c A1 A2 Ak
2 2
A + +L +
( ax + b )
k
2 2 ax + b ax + b ( ax + b ) 2 ( ax + b )
k
ax + b
u 2 a2
ò u 2 - a 2 du = u - a 2 - ln u + u 2 - a 2 + c Ax + B A1 x + B1
+L +
Ak x + Bk
2 2
( ax + bx + c )
2 k
ax 2 + bx + c ax + bx + c ( ax2 + bx + c )
2 k
u 2 a2 æuö ax 2 + bx + c
ò a - u du = a - u + sin -1 ç ÷ + c
2 2 2
2 2 èaø
u-a Products and (some) Quotients of Trig Functions
a2 æ a -u ö
ò 2au - u 2 du =
2
2 au - u 2 + cos-1 ç
2 è a ø
÷+c ò sin x cos x dx
n m
1. If n is odd. Strip one sine out and convert the remaining sines to cosines using
Standard Integration Techniques sin 2 x = 1 - cos2 x , then use the substitution u = cos x
Note that all but the first one of these tend to be taught in a Calculus II class. 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
u Substitution 3. If n and m are both odd. Use either 1. or 2.
ò a f ( g ( x ) ) g¢ ( x ) dx then the substitution u = g ( x ) will convert this into the
b
Given 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.
g ( b)
integral, ò f ( g ( x ) ) g ¢ ( x ) dx = ò ò
b
f ( u ) du . tan n x sec m x dx
a g ( a)
1. If n is odd. Strip one tangent and one secant out and convert the remaining
Integration by Parts tangents to secants using tan 2 x = sec 2 x - 1 , then use the substitution u = sec x
The standard formulas for integration by parts are, 2. If m is even. Strip two secants out and convert the remaining secants to tangents
b b b using sec 2 x = 1 + tan 2 x , then use the substitution u = tan x
ò udv = uv - ò vdu òa udv = uv a - òa vdu 3. If n is odd and m is even. Use either 1. or 2.
Choose u and dv and then compute du by differentiating u and compute v by using the 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
fact that v = ò dv .
Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins