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Common Derivatives and Integrals
Common Derivatives and Integrals
Derivatives Basic Properties/Formulas/Rules Properties/Formulas/Rules d ( cf ( x ) ) = cf ¢ ( x ) , c is any constant. dx d n ( x ) = nx n -1 , n is any number. dx
( f ( x ) ± g ( x ) )¢ = f ¢ ( x ) ± g¢ ( x ) d dx
b
( c ) = 0 , c is any constant.
d d
( f ( g ( x)) ) = f ¢ ( g ( x )) g¢ ( x)
(Chain Rule)
d
(e ( ) ) = g ¢( x ) e ( ) dx g x
( ln g ( x )) = dx
g x
Common Derivatives Polynomials d d (c) = 0 ( x ) = 1 dx dx
Trig Functions d ( sin x ) = cos x dx d ( sec x ) = sec x tan x dx Inverse Trig Functions d 1 ( sin -1 x ) = dx 1 - x2 d 1 ( sec -1 x ) = dx x x2 -1
d dx
d dx d dx
( cx ) = c
d
g ¢ ( x )
n
d
( cos x ) = - sin x ( csc x ) = - csc x cot x
dx d dx
b
b
b
b
a
a
a
a
a
a
b
a
a
a
b
b
c
b
b
a
a
c
a
b
d
( cx ) = ncx dx
n 1
a
b
a
g ( x)
( x ) = nx dx
b
a
æ f ö¢ f ¢ g - f g ¢ – (Quotient Rule) ( f g )¢ = f ¢ g + f g ¢ – (Product Rule) ç ÷ = g 2 è g ø dx
ò cf ( x ) dx = c ò f ( x ) dx , c is a constant. ò ( x ) ± g ( x ) dx = ò f ( x) dx ± ò g ( x) dx ò ( x ) dx = F ( x) = F ( b ) - F ( a ) where F ( x ) = ò f ( x ) dx 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 ò f ( x) dx = 0 ò f ( x ) dx = - ò f ( x ) dx dx = c ( b - a ) ò ( x ) dx = ò f ( x ) dx + ò f ( x) dx ò c dx If f ( x ) ³ 0 on a £ x £ b then ò f ( x ) dx ³ 0 If f ( x ) ³ g ( x ) on a £ x £ b then ò f ( x ) dx ³ ò g ( x ) dx
n 1
n
b
b
a
a
Common Integrals Polynomials
1
ò dx = x + c
dx = k x + c ò k dx
ò x dx = n + 1 x
( tan x ) = sec2 x
1 ó ô dx = ln x + c õ x
ò x
ò x
( cot x ) = - csc
1 1 ó dx = ln ax + b + c ô õ ax + b a
2
x
-1
n
dx = ln x + c
ò x
p q
dx =
-n
dx =
1 p q
+1
p
xq
n+1
1
-n + 1
+1
+ c=
x
+ c, n ¹ - 1 - n +1
+ c, n ¹ 1
q p + q
x
p +q q
+c
Trig Functions d dx d dx
-1
1
-1
1- x 1
( cos x ) = ( csc x ) = -
x
d 2
x2 - 1
dx d dx
( tan
-1
x ) =
ò cos u du = sin u + c ò sin u du = - cos u + c ò sec u du = tan u + c udu = - csc u + c ò sec u tan u du = sec u + c ò csc u cot ud 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
1
2
1+ x
2
2
1
-1
( cot x ) = - 1 + x
2
3
Exponential/Logarithm Functions d x d x ( a ) = a x ln ( a ) (e ) = ex dx dx d d 1 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
ó 1 -1 æ u ö ô 2 2 du = sin ç a ÷ + c è ø õ a -u ó 1 du 1 -1 æ u ö c ô 2 2 = tan ç ÷ + a õ a +u è aø 1 1 ó æuö du = sec-1 ç ÷ + c ô 2 2 a èaø õ u u -a
ò sin
-1
ò tan
u du = u sin -1 u + 1 - u 2 + c
-1
ò cos
-
u du = u tan 1 u -
1 2
ln (1+ u
2
)+ c
-1
u du = u cos - 1 u - 1 - u 2 + c
Hyperbolic Trig Functions
ò sinh u du = cosh u + c ò cosh u du = sinh u + c ò sech ò sech tanh u du = - sech u + c ò csch coth u du = - csch u + c ò csch ò tanh u du = ln ( cosh u ) + c ò sech u du = tan sinh u + c
2
u du = tanh u + c
2
u du = - cot h u + c
ò
a + u du =
ò
u - a du =
2
2
2
2
u 2 u 2 u
ò
a 2 - u 2 du =
ò
2 2au - u du =
2
a +u + 2
2
u -a 2
2
a2 - u 2 + u-a 2
a2 2 2 a 2 a2
u-a 1 1 ó +c ô 2 2 du = ln õ u -a 2a u + a ln u + ln u +
2
a2 2
2
ax 2 + bx + c
ò sin x cos n
b
a
g b
g a
Integration by Parts The standard formulas for integration by parts are,
ò udv = uv - ò vdu
ò
b a
Ax + B ax + bx + c 2
( ax
2
Term in P.F.D A1
k
+ bx + c )
ax + b
+
A2
( ax + b )
A1 x + B1
k
ax2 + bx + c
2
+L +
+L +
Ak
( ax + b )
k
Ak x + Bk
( ax 2 + bx + c )
k
Products and (some) Quotients of Trig Functions Functions
æ a-uö ÷+ c è a ø
cos ç
ò f ( g ( x) ) g¢ ( x) dx then the substitution u = g ( x ) will convert this into the ( ) f ( u ) du . integral, ò f ( g ( x) ) g ¢ ( x ) dx = ò ( ) b
( ax + b )
ax + b
u -a +c 2
-1
Term in P.F.D Factor in Q ( x ) A
ax + b
u Substitution
a
Factor in Q ( x )
2
Standard Integration Techniques Note that all but the first one of these tend to be taught in a Calculus II class.
Given
fraction decomposition decomposition of the rational expression. Integrate the partial fraction decomposition (P.F.D.). For each factor in the denominator we get get term(s) in the decomposition decomposition according to the following table.
a +u + c 2
æuö sin -1 ç ÷ + c 2 è aø
2 au - u +
ó P ( x ) dx where the degree (largest exponent) of P ( x ) is smaller than the If integrating ô õ Q (x) degree of Q ( x ) then factor the denominator as completely as possible and find the partial
-1
Miscellaneous u+a 1 1 ó +c ô 2 2 du = ln õ a -u 2a u - a
Trig Substitutions If the integral contains th e following root use the given substitution and formula. a a 2 - b2 x 2 x = sin q Þ and cos 2 q = 1 - sin 2 q b a b2x2 - a2 x = sec q Þ and tan 2 q = sec 2 q - 1 b a 2 2 a 2 + b2 x2 x = tan q and sec q = 1 + tan q Þ b Partial Fractions
b
udv = uv a -
ò
b
a
vdu
Choose u and dv and then compute du by differentiating u and compute v by using the
ò
m
x dx
1. If n is odd. Strip one sine out and convert the remaining sines to cosines using 2 2 si n x = 1 - cos x , then use the substitution u = cos x 2. If m is odd. Strip one cosine out and convert the remaining cosines to sines 2 2 using cos x = 1 - sin 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 formulas to reduce the integral into a form that can be integrated.
ò tan x sec n
m
x dx
1. If n is odd. Strip one tangent and one secant out a nd 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 ta ngents 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. differently. Convert Example : cos 6 x = ( cos 2 x )
