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Title: Derivatives and Integrals
Description: Here is a set of common derivatives and integrals that are used somewhat regularly in a Calculus I or Calculus II class. Also included are reminders on several integration techniques. Currently this cheat sheet is four pages long.
Description: Here is a set of common derivatives and integrals that are used somewhat regularly in a Calculus I or Calculus II class. Also included are reminders on several integration techniques. Currently this cheat sheet is four pages long.
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Common Derivatives and Integrals
Derivatives
Basic Properties/Formulas/Rules
d
( cf ( x ) ) = cf ′ ( x ) , c is any constant
...
( c ) = 0 , c is any constant
...
math
...
edu for a complete set of Calculus I & II notes
...
) dx
∫ a f ( x=
b
F (=
x) a
b
∫ f ( x ) ± g ( x ) dx = ∫ f ( x ) dx ± ∫ g ( x ) dx
F ( b ) − F ( a ) where F ( x ) = ∫ f ( x ) dx
∫ a cf ( x ) dx = c ∫ a f ( x ) dx , c is a constant
...
math
...
edu for a complete set of Calculus I & II notes
...
u Substitution
∫ a f ( g ( x ) ) g ′ ( x ) dx then the substitution u = g ( x ) will convert this into the
b
g (b)
integral, ∫ f ( g ( x ) ) g ′ ( x ) dx = ∫
f ( u ) du
...
Visit http://tutorial
...
lamar
...
© 2005 Paul Dawkins
Common Derivatives and Integrals
Trig Substitutions
If the integral contains the following root use the given substitution and formula
...
Integrate the partial fraction
decomposition (P
...
D
...
Factor in Q ( x )
Term in P
...
D Factor in Q ( x )
ax + b
A
ax + b
( ax + b )
ax 2 + bx + c
Ax + B
2
ax + bx + c
( ax 2 + bx + c )
Term in P
...
D
Ak
A1
A2
+
+ +
2
k
ax + b ( ax + b )
( ax + b )
k
Ak x + Bk
A1 x + B1
+ +
k
2
ax + bx + c
( ax 2 + bx + c )
k
Products and (some) Quotients of Trig Functions
n
m
∫ sin x cos x dx
1
...
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
...
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
...
Use either 1
...
4
...
Use double angle formula for sine and/or half angle
formulas to reduce the integral into a form that can be integrated
...
If n is odd
...
If m is even
...
If n is odd and m is even
...
or 2
...
If n is even and m is odd
...
Convert Example : cos 6 x=
( cos x ) = (1 − sin x )
2
3
2
3
Visit http://tutorial
...
lamar
...
© 2005 Paul Dawkins
Title: Derivatives and Integrals
Description: Here is a set of common derivatives and integrals that are used somewhat regularly in a Calculus I or Calculus II class. Also included are reminders on several integration techniques. Currently this cheat sheet is four pages long.
Description: Here is a set of common derivatives and integrals that are used somewhat regularly in a Calculus I or Calculus II class. Also included are reminders on several integration techniques. Currently this cheat sheet is four pages long.