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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
−
−
= +
∫
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