5.c – The Fundamental Theorem of
Calculus and Definite Integrals
d
dx

x
a
f  t  dt


b
a
f  x  dx
Examples
The definite integral of f (x) from x = a to x = b is
denoted

b
a
f  x  dx
f(x) is called the integrand, a the lower limit of
integration, and b the upper limit of integration.
The First Fundamental Theorem of
Calculus
If F  x    f ( x ) dx be the general antiderivative
of f  x  , then
 f  x  dx  F b   F  a   F  x    F ( x)
Calculator : fnInt  f  x  , x, a, b 
b
b
b
a
a
a
 MATH | 9 or ALPHA | F2 on newer models
Basic Properties of the Indefinite Integral
Let a, b, and c be constants and f and g be
continuous functions on [a, b].
1.
2.
3.
4.
 f  x  dx   f  x  dx
a
b
b
a
 f  x   g  x  dx   f  x  dx 
a
a

b

b
a
b
 g  x  dx
b
a
c f  x  dx  c  f  x  dx
b
a
 f  x  dx   f  x  dx   f  x  dx;
c
b
c
a
a
b
a bc
4
Examples
Evaluate:
1.

8
1
2. 

 /4
0

2  3 x 2 dx
sec t tan t dt
1
3.  10 x dx
0
4. 
2
2
if  2  x  0
2
f  x  dx where f  x   
2
4

x
if 0  x  2

Definite Integrals With The Substitution
Rule
If u = g(x) is a differentiable function whose range is an
interval I and f is continuous on I, then

b
a
f  g  x   g   x  dx  
g (b )
g (a)
f u  du
Properties of Odd and Even: Suppose f is continuous on
[– a, a].
(a) If f is even  f   x   f  x   , then  f ( x)dx  2  f ( x)dx.
a
0
a
a
(b) If f is odd  f   x    f  x   , then  f ( x)dx  0.
a
a
6
Examples - Evaluate
Evaluate by changing your limits of integration to
values that are in terms of u.

4
x
2
5. 
x cos  x  dx
8. 
dx
0
0
1 2x
1
1
1/ 2 sin
2
x
x
6.  xe dx
9. 
dx
0
0
1  x2
2
 / 2 x sin x
7. 
dx
6
 / 2 1  x
7
First Fundamental Theorem of Calculus
(Alternate Definition)
We’ve shown that  f  x  dx represents the general
antiderivative of f with respect to x. It follows that the
derivative of the antiderivative should return the original
function (that is, the integrand).
d
dx

Upper Limit Must
Be Variable Part
x
a

d
f  t  dt   F ( x )  F (a )  f  x 
dx
Lower Limit Must
Be Numerical Part
t is called a dummy variable.
Examples
Evaluate the following derivatives.
d 1 3
11.  t dt
dx x
d x
10.
cos t dt

dx 1
d
13.
dx

x
1
ln t dt

d  x 3

15.
dt


dx  3 t 2  5t 
d x 1
12.
dt
2

dx 2 1  t

d
16.
dx  
d
14.
du
u
1
2
x

9 csc t dt 
cos  t 2  dt
Examples
Let u is some function of x. Use WolframAlpha
to determine the following:
d sin  x  5
17.
r dr

dx 1
d 2
18.
cos  t  dt
4

dx x 3 x
d u
f  t  dt  derivative integral  f (t ), t , a, u , x 

dx a
First Fundamental Theorem of
(Generalized)
If f is continuous on [a, b] and u is an unknown,
differentiable function of x, then
d
dx

u
a

f  t  dt  f  ___   _____
Examples
Evaluate
d  e2 x

19.
  2  t  sin t  dt 
dx 

d
20.
dx

1
5


sec
t

t
dt
ln x 

