DERIVATIVES
7.1: Antiderivatives
Objectives:
•To find the antiderivative of a function using the rules of
antidifferentiation
•To find the indefinite integral
•To apply the indefinite integral
• Up until this point, we have done problems
such as: f(x)= 2x +7, find f’(x)
• Now, we are doing to do problems such as:
f’(x)=2, find f(x)
• We do this through a process called
antidifferentiation
Warm Up:
1. Find a function that has the derivative
f’(x)=3x2+2x
2. Find a function that has the derivative
f’(x)=x4-4x3+2
DEFINITION: ANTIDERIVATIVE
If F’(x) = f(x) then F(x) is an antiderivative of f(x)
F’(x) = 2x, then F(x) = x2 is the antiderivative of
2x (it is a function whose derivative is 2x)
Find an antiderivative of 6x5
2 antiderivatives of a function can differ only by
a constant:
f’(x) = 2x
F(x) = x2 +3
g’(x)=2x
G(x)=x2-1
F(x)-G(x)= C
The constant, C, is called an integration constant
INDEFINITE INTEGRAL!!!!!


f ( x) dx
integral sign
f(x)
integrand
dx
change in x (remember differentials?!?!)
Be aware of variables of integration…
If F’(x) = f(x), then
real number C
 f ( x)dx = F(x) + C, for any
F(x) is the antiderivative of f(x)
This is a big deal!!!!!!!
Example:
Find the indefinite integral.
2
xdx

Rules of Integration
Power Rule
Constant Multiple Rule
Sum or Difference Rule
n 1
x
n
x
 dx  n  1
 k  f ( x)dx  k  f ( x)dx
(k has to be a real #, not a variable)
  f ( x)  g ( x)dx   f ( x)dx   g ( x)dx
Examples: Find the indefinite integral
1.
 2 x  3x dx
2
2.
 cos xdx
3.
 4t dt
3
2
x
2
4.
 x dx
5.
 x
2

2
 5 dx
More Rules…..
 e dx  e
x
x
C
kx
e
kx
e
 dx  k  C
1
 x dx   x dx  ln x  C
1
Examples: Find the Indefinite Integral
6
1. dx
x


2. e3 x  x 2 dx
1
t
 2
2
3.  sec t  e dt


Initial Value Problems
Find the function, f(x), that has the following:
1
f ' ( x)  2  x; f (2)  1
x
f ' ( x)  3x  1; f (2)  4
2
Find an equation of the curve whose tangent line has a slope of
f’(x)=x2/3 given the point (1, 3/5) is on the curve.
Applications
1. An emu is traveling on a straight road. Its
acceleration at time t is given by a(t)=6t+4 m/hr2.
Suppose the emu starts at a velocity of -6 mph
(crazy…its moving backwards) at a position of 9
miles. Find the position of the emu at any time, t.
(Acceleration due to gravity= -32 ft/sec2)
A stone is dropped from a 100 ft building. Find, as a
function of time, its position and velocity. When does
it hit the ground, and how fast is it going at that time?