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MATH 275 Summary Of Common Derivatives And Integrals

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University of Calgary

Calculus for Engineers and Scientists (MATH 275)

Calculus Cheat Sheet Integrals

Calculus Cheat Sheet
Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins
Integrals
Definitions
Definite Integral: Suppose f x( ) is continuous
on [ ] a b, . Divide [ ] a b, into n subintervals of
width ∆ x and choose *
i
x from each interval.
Then ( ) ( ) *
1
lim i
n
b
a n
i
f x dx f x x
→∞
=

= ∆ ∑ .
Anti-Derivative : An anti-derivative of f x( )
is a function, F x( ) , such that Fx fx ′( ) ( ) = .
Indefinite Integral : f x dx F x c ( ) ( ) = + ∫
where F x( ) is an anti-derivative of f x( ).
Fundamental Theorem of Calculus
Part I : If f x( ) is continuous on [ ] a b, then
( ) ( ) x
a
g x f t dt = ∫
is also continuous on [ ] a b,
and ( ) ( ) ( ) x
a
d
g x f t dt f x
dx
′ = = ∫
.
Part II : f x( )is continuous on[ ] a b, , F x( ) is
an anti-derivative of f x( )(i.e. F x f x dx ( ) ( ) = ∫
)
then ( ) ( ) ( ) b
a
f x dx F b F a = − ∫
.
Variants of Part I :
u x
a
d
f t dt u x f u x
dx v x
d
f t dt v x f v x
dx v x
ux vx
d
f t dt u x f v

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MATH 275 Summary Of Common Derivatives And Integrals

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