AP微积分BC公式大全

b
If f ( x ) ³ g ( x ) on a £ x £ b then Common Integrals Polynomials
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ò dx = x + c
ó 1 dx = ln x + c ô õx
ò k dx = k x + c òx
-1
ò x dx = n + 1 x
n
1
n +1
+ c, n ¹ -1
ò sin
-1
u du = u sin -1 u + 1 - u 2 + c 1 u du = u tan -1 u - ln (1 + u 2 ) + c 2 u du = u cos -1 u - 1 - u 2 + c
ò tan
-1
ò cos
-1
ò sech tanh u du = - sech u + c ò tanh u du = ln ( cosh u ) + c
2 2
2au - u 2 du =
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. u Substitution
au ò e sin ( bu ) du =
+c
ó 1 du = ln ln u + c ô õ u ln u
© 2005 Paul Dawkins
Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes.
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)
d ( tan x ) = sec2 x dx d ( cot x ) = - csc2 x dx d 1 tan -1 x ) = ( dx 1 + x2 d 1 cot -1 x ) = ( dx 1 + x2
Exponential/Logarithm Functions d x d x a ) = a x ln ( a ) e ) = ex ( ( dx dx d 1 d 1 ln ( x ) ) = , x > 0 ln x ) = , x ¹ 0 ( ( dx x dx x Hyperbolic Trig Functions d ( sinh x ) = cosh x dx d ( sech x ) = - sech x tanh x dx d ( cosh x ) = sinh x dx d ( csch x ) = - csch x coth x dx
© 2005 Paul Dawkins
Common Derivatives and Integrals
Integrals
Basic Properties/Formulas/Rules ò cf ( x ) dx = c ò f ( x ) dx , c is a constant.
b a b
ò f ( x ) ± g ( x ) dx = ò f ( x ) dx ± ò g ( x ) dx b b ò a f ( x ) dx = F ( x ) a = F ( b ) - F ( a ) where F ( x ) = ò f ( x ) dx
+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
cf ( x ) dx = c ò f ( x ) dx , c is a constant.
a
òa òa
òa
b b
f ( x ) ± g ( x ) dx = ò f ( x ) dx ± ò g ( x ) dx
a a a
b
b
ò a f ( x ) dx = 0
b
ò a f ( x ) dx = -òb f ( x ) dx
2 2 3
ò csc u du = ln csc u - cot u + c
Exponential/Logarithm Functions
ò csc
u
3
u du =
1 ( - csc u cot u + ln csc u - cot u ) + c 2
òe
u
du = e + c
u
au ò a du = ln a + c
© 2005 Paul Dawkins
Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes.
Common Derivatives and Integrals
Trig Substitutions If the integral contains the following root use the given substitution and formula. a a2 - b2 x2 x = sin q Þ and cos 2 q = 1 - sin 2 q b a b2 x2 - a2 Þ x = sec q and tan 2 q = sec2 q - 1 b a a2 + b2 x2 Þ x = tan q and sec 2 q = 1 + tan 2 q b Partial Fractions ó P ( x) dx where the degree (largest exponent) of P ( x ) is smaller than the If integrating ô õ Q ( x)
Common Derivatives and Integrals
Inverse Trig Functions 1 ó æuö du = sin -1 ç ÷ + c ô èaø õ a2 - u2 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 2 2 a èaø õ u u -a Hyperbolic Trig Functions ò sinh u du = cosh u + c
dx = ln x + c
p
òx
q ò x dx =
-n
dx =
p
1 x - n +1 + c, n ¹ 1 -n + 1
p+q q
ó 1 dx = 1 ln ax + b + c ô õ ax + b a Trig Functions ò cos u du = sin u + c
q 1 q +1 x +c = x p p+q q +1
d ( cx ) = c dx
d n ( x ) = nxn-1 dx
d ( cx n ) = ncxn-1 dx
d ( cos x ) = - sin x dx d ( csc x ) = - csc x cot x dx d 1 cos -1 x ) = ( dx 1 - x2 d 1 csc -1 x ) = ( dx x x2 -1
d 1 log a ( x ) ) = , x>0 ( dx x ln a d ( tanh x ) = sech 2 x dx d ( coth x ) = - csch 2 x dx
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
e au ( a sin ( bu ) - b cos ( bu ) ) + c a2 + b2 eau au e cos bu du = ( ) ( a cos ( bu ) + b sin ( bu ) ) + c ò a 2 + b2
Given
b
Integration by Parts The standard formulas for integration by parts are,
ò udv = uv - ò vdu
òa
b
udv = uv a - ò vdu
a
b
b
Choose u and dv and then compute du by differentiating u and compute v by using the fact that v = ò dv .
Common Derivatives and Integrals
Derivatives
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 x ) = nx n-1 , n is any number. ( c ) = 0 , c is any constant. ( 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 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
Miscellaneous ó 1 du = 1 ln u + a + c ô 2 õ a - u2 2a u - a
ò cosh u du = sinh u + c ò sech ò csch coth u du = - csch u + c ò csch ò sech u du = tan sinh u + c
c b c
f ( x ) dx = ò f ( x ) dx + ò f ( x ) dx
a b
ò a c dx = c ( b - a )
f ( x ) dx ³ ò g ( x ) dx
a b
b
If f ( x ) ³ 0 on a £ x £ b then
ò a f ( x ) dx ³ 0 òa
-1
2 2
u du = tanh u + c u du = - coth u + c
ó 1 du = 1 ln u - a + c ô 2 õ u - a2 2a u + a
ò ò ò ò
u 2 a2 2 a + u du = a + u + ln u + a 2 + u 2 + c 2 2 u 2 a2 u 2 - a 2 du = u - a 2 - ln u + u 2 - a 2 + c 2 2 u 2 a2 æuö a 2 - u 2 du = a - u 2 + sin -1 ç ÷ + c 2 2 èaø