# 5.3 â€“ The Fundamental Theorem of Calculus

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```					5.3 – The Fundamental Theorem of
Calculus
Examples
Use the TI-89 to determine g.
2. g  x    t 3 dt
x
1. g  x    cos t dt
x

1                                  3

1
3. g  x   
x
dt
1    1 t 2

Question So what does g represent here?
Answer A specific antiderivative of the function f.
First Fundamental Theorem of
Calculus – Part 1
If f is continuous on [a, b], then the function g is
defined by

g  x    f  t  dt
x

a

is continuous on [a, b] and differentiable on
(a, b), and g'(x) = f (x). That is, g is the
antiderivative of f in terms of x.
First Fundamental Theorem of Calculus
– Part 1 (Alternate Definition)

If f is continuous on [a, b], then

d  x
f t  dt   f x 
dx  a
              

Examples

3
1. g  x    ln t dt       2. g  x   
x                               x
dt
1                            3       t  5t
2

3. g  x    cos  t  dt 4. g  x   
u                               2
2
9csc t dt
1                                x
Examples
Use the TI-89 to determine the derivative of g.
Establish a property for what you observe.
d x2
dx 1
5.       1  r 3 dr

d cos x
6.
dx 3 ln t dt
First Fundamental Theorem of Calculus
– Part 1 (Alternate Definition)

If f is continuous on [a, b] and u is a function of
x, then

d  u               f u  du
dx 
 a f t  dt 
          dx
Examples

Determine the derivative of g by hand.
e2 x
7. g  x               t  sin t  dt
2

8. g  x    sin t dt
2
5
ln x
First Fundamental Theorem of
Calculus – Part 2
If f is continuous on [a, b], then the function g is
defined by

 f  x  dx  F  b   F  a 
b

a

where F is the general antiderivative of f,
that is, a function such that F ' = f.

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