# Calculus Cheat Sheet Integrals

No need to memorize the Derivatives and Integral of Trigonometric Functions with this Technique
No need to memorize the Derivatives and Integral of Trigonometric Functions with this Technique

Calculus Cheat Sheet Integrals

University: Queensland University of Technology

Course: Calculus and Linear Algebra 2 (MATH2000)

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Calculus Cheat Sheet

Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins

Integrals

Definitions

Definite Integral: Suppose

( )

fx

is continuous

on

[ ]

,ab

. Divide

[ ]

,ab

into n subintervals of

width

x∆

and choose

*

i

x

from each interval.

Then

( )

( )

*

1

lim

i

n

b

ani

f x dx f x x

→∞ =

= ∆

.

Anti-Derivative : An anti-derivative of

( )

fx

is a function,

( )

Fx

, such that

( ) ( )

Fx fx

′=

.

Indefinite Integral :

( ) ( )

f x dx F x c= +

where

( )

Fx

is an anti-derivative of

( )

fx

.

Fundamental Theorem of Calculus

Part I : If

( )

fx

is continuous on

[ ]

,ab

then

( ) ( )

x

a

g x f t dt=∫

is also continuous on

[ ]

,ab

and

( ) ( ) ( )

x

a

d

g x f t dt f x

dx

′= =

.

Part II :

( )

fx

is continuous on

[ ]

,ab

,

( )

Fx

is

an anti-derivative of

( )

fx

(i.e.

( ) ( )

F x f x dx=∫

)

then

( ) ( ) ( )

b

af x dx F b F a= −

.

Variants of Part I :

( )

( )

( ) ( )

ux

a

df t dt u x f u x

dx ′

=

( )

( )

( ) ( )

b

vx

df t dt v x f v x

dx ′

= −

( )

( )

( )

( )

[ ]

( )

[ ]

() ()

ux

vx ux vx

dftdt uxf vxf

dx ′′

= −

Properties

( ) ( ) ( ) ( )

f x g x dx f x dx g x dx±= ±

∫ ∫∫

( ) ( ) ( ) ( )

b bb

a aa

f x g x dx f x dx g x dx±= ±

∫ ∫∫

( )

0

a

af x dx =

( ) ( )

ba

ab

f x dx f x dx= −

∫∫

( ) ( )

cf x dx c f x dx=

∫∫

, c is a constant

( ) ( )

bb

aa

cf x dx c f x dx=

∫∫

, c is a constant

( )

b

a

c dx c b a= −

( ) ( )

bb

aa

f x dx f x dx≤

∫∫

( ) ( ) ( )

b cb

a ac

f x dx f x dx f x dx= +

∫∫∫

for any value of c.

If

( ) ( )

f x gx≥

on

axb≤≤

then

( ) ( )

bb

aa

f x dx g x dx≥

∫∫

If

( )

0fx≥

on

axb≤≤

then

( )

0

b

af x dx ≥

If

( )

m fx M≤≤

on

axb≤≤

then

( ) ( ) ( )

b

a

mba fxdx Mba−≤ ≤ −

Common Integrals

k dx k x c= +

1

1

1,1

nn

n

x dx x c n

+

+

= + ≠−

11ln

x

x dx dx x c

−= = +

∫∫

11

ln

a

ax b

dx ax b c

+

= ++

( )

ln lnu du u u u c= −+

uu

du c= +

∫ee

cos sinu du u c= +

sin cosu du u c=−+

2

sec tanu du u c= +

sec tan secu u du u c= +

csc cot cscu udu u c=−+

2

csc cotu du u c=−+

tan ln secu du u c= +

sec ln sec tanu du u u c= ++

( )

1

11

22

tan

u

aa

au

du c

+

= +

( )

1

22

1

sin

u

a

au

du c

= +