Integration Fact Sheet

Calculus I

Integration Fact Sheet
Properties

Integral Definition

∑ f ( x ) ∆x
n

=
∫ f ( x ) dx
b

lim

max ∆xk →0

a

*
k

k =1

k

∫ c f ( x ) dx = c ∫ f ( x ) dx

∫  f ( x ) ± g ( x ) dx = ∫ f ( x ) dx ± ∫ g ( x ) dx

Net Signed Area

A = ∫ f ( x ) dx
b

a

Constant and Power Rule

total area = ∫ f ( x ) dx
b

∫ 1dx=

a

Mean Value

f (x

f ( x ) dx
∫ =
b

a

f ave

*

r
dx
∫x =

) (b − a )

or

a

b

a

a

b

f ( x ) dx ∫ f ( x ) dx + ∫ f ( x ) dx
∫=
b

c

b

a

a

c

u-sub.

∫

b

a

f ( g ( x ) ) g '( x) dx = ∫

g (b)

g(a)

f ( u ) du

Fundamental Theorem of Calculus
1.

2.

) dx
∫ f ( x=
b

a

( r ≠ −1)
ax3 bx 2
+
+ cx + C
3
2

Logarithmic/Exponential Functions

∫ f ( x ) dx = − ∫ f ( x ) dx

a

x r +1
+C
r +1

2
∫ ( ax + bx + c ) dx =

1 b
=
f ( x ) dx
b − a ∫a
Properties

∫ f ( x ) dx = 0

x+C

F (=
x ) a F (b) − F ( a )
b

where F ' ( x ) = f ( x ) .

d x
g ( t ) dt = g ( x )
dx ∫a
⌠ 1
dx sin −1 x + C
=

2
⌡ 1− x
1
⌠
=
dx sec −1 x + C

2
⌡ x x −1
⌠ 1=
dx tan −1 x + C

⌡ 1 + x2

∫e

x

dx= e x + C

bx
⌠ bx =
+C
 dx
ln b
⌡

⌠ 1=
 dx ln x + C
⌡x

Basic Trigonometric Functions
− cos x + C
∫ sin x dx =
dx sin x + C
∫ cos x=
dx tan x + C
∫ sec x=
2

∫ csc

2

x dx =
− cot x + C

dx
∫ sec x tan x=

sec x + C

− csc x + C
∫ csc x cot x dx =

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