Review




a
a
b
a
b
a
b
a
?( b  a ) 
f ( x )dx 
kf ( x )dx 
( f ( x )  g ( x ))dx 
c
f ( x )dx   f ( x )dx 
b

b
a
f ( x )dx  ??( b  a )
The fundamental Theorem of Calculus
dF
d x

f ( x )dx  ???

dx dx a
d
d x
F ( x) 
f ( x )dx  f ( x )

dx
dx a

x
a
f ( x )dx 

x
a
F ( x )dx  ??
The Mean Value Theorem for Definite Integrals
Indefinite integrals and the substitution
rules***
Substitution in Definite Integrals
n1

d u 
n du

u
dx  n  1 
dx
n1
du
u
 n

  u dx dx  n  1  C
Example 1
2
3
3
x
x
 1 dx


4t  1dt
 f ( g( x ))  g( x )dx
Let u  g ( x ), du  g ( x )dx . Then
 f ( g( x ))  g( x )dx   f ( u)du
 F (u)  C
 F ( g ( x ))  C
Example 3
 cos(7  5)d
2
3
x
sin(
x
)dx

Exercise on P375
Q13, Q14
Example 4
2
3
x
sin(
x
) dx

1
 cos 2 2 x dx

2 zdz
3
z2  1
Example 5
2
sin
 xdx
2
cos
 xdx
Example 7
2
g
(
x
)

sin
x over the
Figure 5.24 shows the graph of
interval [0, 2 ].
Find
(a) the definite integral of g ( x ) over [0, 2 ],
(b)the area between the graph of the function and the
x-axis over [0, 2 ].
Homework on P375
Q20, Q22, Q27, Q28;