465
Answers
701–800
Answers and Explanations
740.
8
9 3
ln
Using the average value formula with a = 0, b = ln 3, and f
 
(x) = sinh x cosh x, gives you
x
f
d x
avg =
− ∫
1
3 0 0
3
ln
sinhcosh
ln
Use the substitution u = cosh x so that du = sinh x. You can find the new limits of integration by noting that if x = ln 3, then u
e
e
=
=
+
=
+ =
−
cosh(ln )
ln
ln
3
2
3 1
3
2
5
3
3
3
, and
if x = 0, then u
e e
=
=
+
=
−
cosh( )
0
2
1
0
0
. With these new values, you have
1
3
1
3 2
1
2 3
5
3
1
8
9 3
1
5 3
2
1
5 3
2
ln
ln
ln
ln
u du
u
∫
=
=
( ) −






=
741.
1
2
Using the average value formula with a = 1, b = 4, and f r
r
( ) (
)
= +
5
1
2
gives you
f
r
dr
r
dr
avg = −
+
=
+
∫
∫
1
4 1
5
1
1
3
5
1
2
1
4
2
1
4
(
)
(
)
Next, use the substitution u = 1 + r so that du = dr. Find the new limits of integration by
noting that if r = 4, then u = 1 + 4 = 5, and if r = 1, then u = 1 + 1 = 2. With these new
values, you find that
1
3
5
5
3
5
3
1
5
3
1
5
1
2
5
3
1
5
1
2
2
2
5
2
2
5
2
5
u
du
u du
u
∫
∫
=
=
−
( )
=
− − −
( )






=
− +
−
( ( )
= ( )
=
5
3
3
10
1
2