5.3 – The Fundamental Theorem of Calculus

Examples

Use WolframAlpha to determine g. Then determine g'(x).

1

1. cosx

g x t dt 21

13.

1

xg x dt

t

Question So what does g represent here?

3

32.

xg x t dt

integral ( ), variable, ,b

af x dx f x a b

First Fundamental Theorem of Calculus – Part 1

If f is continuous on [a, b], then the function g is defined by

x

ag x f t dt

is continuous on [a, b] and differentiable on (a, b), then g(x) is the _______________

of f(t) in terms of x. That is, g'(x) = _____.

First Fundamental Theorem of Calculus – Part 1 (Alternate Definition)

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

______x

a

df t dt

dx

Examples

11. ln

xdt dt

dx 23

32.

5

xddt

dx t t

Evaluate the following.

2

13. cos

udt dt

du 24. 9csc

x

dt dt

dx

Examples

Use WolframAlpha to determine the following:

sin 5

15.

xdr dr

dx 4 3

36. cos

xdt dt

dx

integral{ (deri ), variable, , }vative ,b

af xf x x a xbd

First Fundamental Theorem of Calculus – Part 1 (Alternate Definition)

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

___ _____u

a

df t dt f

dx

Examples

Determine the derivative of g by hand.

2

27. sin

xedt t dt

dx

1 5

ln8. sec

x

dt t dt

dx

First Fundamental Theorem of Calculus – Part 2

If f is continuous on [a, b], then the function g is defined by

( )b b b

aaaf x d F b F ax F x F x

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

Examples

Evaluate:

8 3 2

1

/ 4

0

1

0

2

22

1. 2

2. sec tan

3. 10

2 2 04. where

4 0 2

x

x dx

t t dt

dx

if xf x dx f x

x if x