Antiderivatives
Antiderivatives
• Mr. Baird knows the velocity of particle and wants
to know its position at a given time
• Ms. Bertsos knows the rate a population of
bacteria is increasing and she wants to know what
the size of the population will be at a future time.
• In each case the rate of change (the derivative) is
known….but what is the original function?
• The original function is called the
ANTIDERIVATIVE of the rate of change.
DEFINITION
A function
F
is called an
antiderivative of
interval
I
for all x in
on an
F x  f  x
if
I
f
.
Suppose f  x   2 x
What is its antiderivative?
We can make some guesses
x 2
2
x 3
2
x
2
x  100000000
2
They all fit!
Theorem
If
F
is an antiderivative of
an interval
I
on
, then the most
general antiderivative of
f
on
I
F  x  C
is
where
f
C
is an arbitrary constant.
Finding an antiderivative is also known as Indefinite
Integration and the Antiderivative is the Indefinite
Integral (Especially for us old guys!)
And the symbol for integration is an elongated S

More on why it’s an S later!
Integrand

Constant of Integration
f  x  dx  F  x   C
Variable of Integration
This is read: The antiderivative of f with respect to x or the
indefinite integral of f with respect to x is equal to…..
What is the Antiderivative of
cos x ?
dy
 cos x
dx
Derivative
dy  cos x dx
We “kinda” multiply
Take the integral of
both sides
 dy   cos x dx
We know what to differentiate
to get cos x
y  sin x  C
F  x
Some General Rules
They are just the derivative rules in reverse
Differentiation Formula
Integration Formula
d
C   0
dx
d
 kx   k
dx
 0 dx  C
 k dx  kx  C
d
k f  x  k f  x
dx
 k f  x  dx  k  f  x  dx
“Pulling out a konstant”
Some General Rules
Differentiation Formula
Integration Formula
d
f  x   g  x   f  x   g x 

dx
  f  x   g  x  dx   f  x  dx   g  x  dx
Sum / Difference Rule for Integrals
 
d n
x  nx n1
dx
n 1
x
 x dx  n  1  C
n
Power Rule for Integrals
Some General Rules
Differentiation Formula
d
 sin x   cos x
dx
d
 cos x    sin x
dx
Integration Formula
 cos x dx  sin x  C
 sin x dx   cos x  C
All the other trig functions follow