Integral

Integral = Antiderivatives

Derivative;

Integral;

( ) ( )F x f x→

( ) ( )f x F x→

Konsep limit dan slope

( )F x( ) ( )

0limx

y dyF x f x

x dx∆ →

∆′= = =

∆

y∆

x∆ x

Konsep limit dan

area di bawah kurva

Y=f(x)

Y

x

∫ dxxf )(

Definite (proper) integral

Y=f(x)

Y

x

∫b

a

dxxf )(

a b

Integral vs derivative

( )

( ) ( )

y F x

dyy F x f x

dx

=

′ ′= = =( ) ( )

( )

( ) ( )

y F x f xdx

dy f x dx

dy f x dx F x c

′ ′= = =

=

= = +∫ ∫

Aturan pangkat (Power rule)

( ) 31

3y F x x= =

( ) 2′ = = ( ) n′ = =( ) 2y f x x′ = =

2 31

3x x=∫

( ) ny f x x′ = =

( )11

1

n nx x

n

+=+∫

Aturan Penambahan/Pengurangan

( ) ( ) ( ) ( )f x g x dx f x dx g x dx ± = ± ∫ ∫ ∫

( ) ( ) ( ) ( ) ( )1 2 1 2F x c G x c F x G x c c= + ± + = ± + +

( ) ( )F x G x c= + +

Definite integral

� = lower limit

( ) ( ) ( )b

a

f x dx F b F a= −∫

a� = lower limit

� = upper limit

( ) ( )5 5

3 32 3

11

3 5 1 125 1 124x dx x = = − = − =∫

b

Improper integral

( )a

f x dx

∞

∫ ( )lim

b

ba

f x dx→∞ ∫

a

( )b

f x dx−∞

∫

a

( )lim

b

aa

f x dx→∞ ∫