5.3 Fundamental Theorem of Calculus
Morro Rock, California
Photo by Vickie Kelly, 1998
Greg Kelly, Hanford High School, Richland, Washington
The Fundamental Theorem of Calculus, Part 1


If f is continuous on a, b , then the function
F  x    f  t  dt
x
a


has a derivative at every point in a, b , and
dF d x

f  t  dt  f  x 

dx dx a

First Fundamental Theorem:
d x
f
t
dt

f
x





dx a
1. Derivative of an integral.

First Fundamental Theorem:
d x
f
t
dt

f
x





dx a
1. Derivative of an integral.
2. Derivative matches upper limit of integration.

First Fundamental Theorem:
d x
f
t
dt

f
x





dx a
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.

First Fundamental Theorem:
d x
f
t
dt

f
x





dx a
New variable.
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.

The long way:
First Fundamental Theorem:
d x

cos
x
cos
t
dt
dx 

d
x
sin t 
dx
d
dx

0
 sin x  sin    
1. Derivative of an integral.
2. Derivative matches
upper limit of integration.
3. Lower limit of integration
is a constant.
d
sin x
dx
cos x

d x 1
1
dt 
2

dx 0 1+t
1  x2
1. Derivative of an integral.
2. Derivative matches
upper limit of integration.
3. Lower limit of integration
is a constant.

d x
cos t dt

dx 0
2
The upper limit of integration does
not match the derivative, but we
could use the chain rule.
 
d 2
cos x  x
dx
2
 
cos x 2  2 x
 
2 x cos x 2

d 5
3t sin t dt

dx x
The lower limit of integration is not
a constant, but the upper limit is.
We can change the sign of the
integral and reverse the limits.
d x
  3t sin t dt
dx 5
3x sin x

d x 1
dt
t

dx 2 x 2  e
2
Neither limit of integration is a
constant.
We split the integral into two parts.
0
d  x2 1
1

dt  
dt 
 0
t
t
2x 2  e
dx  2  e

It does not
matter what
constant we use!
2x
d  x2 1
1

dt  
dt 
 0
t
t
0 2e
dx  2  e

(Limits are reversed.)
1
1
2x
2
 2x 
 2 (Chain xrule
2  is used.)
2x
2x
x2
2e
2e
2e
2e

The Fundamental Theorem of Calculus, Part 2


If f is continuous at every point of a, b , and if
F is any antiderivative of f on  a, b , then
 f  x  dx  F b   F  a 
b
a
(Also called the Integral Evaluation Theorem)
We already know this!
To evaluate an integral, take the anti-derivatives and subtract.
