Second Fundamental Theorem of Calculus
Morro Rock, California
Photo by Vickie Kelly, 1998
Greg Kelly, Hanford High School, Richland, Washington
Here is my favorite calculus textbook quote of all time,
from CALCULUS by Ross L. Finney and George B.
Thomas, Jr., ©1990.
If you were being sent to a desert island
and could take only one equation with you,
d x
f  t  dt  f  x 

dx a
might well be your choice.

The Second Fundamental Theorem of Calculus


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

Second Fundamental Theorem:
d x
f
t
dt

f
x





dx a
1. Derivative of an integral.

Second Fundamental Theorem:
d x
f
t
dt

f
x





dx a
1. Derivative of an integral.
2. Derivative matches upper limit of integration.

Second 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.

Second 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:
Second 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 
t
t
2x 2  e
dx  0 2  e

It does not
matter what
constant we use!
2x
d  x2 1
1
(Limits are reversed.)
 
dt  
dt 
t
t
0 2e
dx  0 2  e

1
1

 2x 
2
2x
x2
2e
2e
2x
2
(Chain xrule
2  is used.)
2x
2e
2e

The First Fundamental Theorem of Calculus


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
We already know this!
To evaluate an integral, take the anti-derivatives and subtract.
