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1. 5.4 Fundamental Theorem of Calculus
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998
Morro Rock, California

2. If you were being sent to a desert island
and could take only one equation with you,
( ) ( )
x
a
d
f t dt f x
dx
=∫
might well be your choice.
Here is my favorite calculus textbook quote of all time,
from CALCULUS by Ross L. Finney and George B.
Thomas, Jr., ©1990.
→

3. The Fundamental Theorem of Calculus, Part 1
If f is continuous on , then the function[ ],a b
( ) ( )
x
a
F x f t dt= ∫
has a derivative at every point in , and[ ],a b
( ) ( )
x
a
dF d
f t dt f x
dx dx
= =∫
→

4. ( ) ( )
x
a
d
f t dt f x
dx
=∫
First Fundamental Theorem:
1. Derivative of an integral.
→

5. ( ) ( )a
xd
f t dt
x
f x
d
=∫
2. Derivative matches upper limit of integration.
First Fundamental Theorem:
1. Derivative of an integral.
→

6. ( ) ( )a
xd
f t dt f x
dx
=∫
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
First Fundamental Theorem:
→

7. ( ) ( )
x
a
d
f t dt f x
dx
=∫
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
New variable.
First Fundamental Theorem:
→

8. cos
xd
t dt
dx π−∫ cos x= 1. Derivative of an integral.
2. Derivative matches
upper limit of integration.
3. Lower limit of integration
is a constant.
( )sin
xd
t
dx π−
( )( )sin sin
d
x
dx
π− −
0
sin
d
x
dx
cos x
The long way:
First Fundamental Theorem:
→

9. 20
1
1+t
xd
dt
dx ∫ 2
1
1 x
=
+
1. Derivative of an integral.
2. Derivative matches
upper limit of integration.
3. Lower limit of integration
is a constant.
→

10. 2
0
cos
xd
t dt
dx ∫
( )2 2
cos
d
x x
dx
⋅
( )2
cos 2x x⋅
( )2
2 cosx x
The upper limit of integration does
not match the derivative, but we
could use the chain rule.
→

11. 5
3 sin
x
d
t t dt
dx ∫
The lower limit of integration is not
a constant, but the upper limit is.
5
3 sin
xd
t t dt
dx
− ∫
3 sinx x−
We can change the sign of the
integral and reverse the limits.
→

12. 2
2
1
2
x
tx
d
dt
dx e+∫
Neither limit of integration is a
constant.
2
0
0 2
1 1
2 2
x
t tx
d
dt dt
dx e e
 
+ 
+ + 
∫ ∫
It does not
matter what
constant we use!
2
2
0 0
1 1
2 2
x x
t t
d
dt dt
dx e e
 
− 
+ + 
∫ ∫
2 2
1 1
2 2
22
xx
x
ee
⋅ − ⋅
++
(Limits are reversed.)
(Chain rule is used.)2 2
2 2
22
xx
x
ee
= −
++
We split the integral into two parts.
→

13. The Fundamental Theorem of Calculus, Part 2
If f is continuous at every point of , and if
F is any antiderivative of f on , then
[ ],a b
( ) ( ) ( )
b
a
f x dx F b F a= −∫
[ ],a b
(Also called the Integral Evaluation Theorem)
We already know this!
To evaluate an integral, take the anti-derivatives and subtract.
π