Part II: The Answers
464
Answers
701–800
738.
1
2π
Using the average value formula with a = 0, b = π
2
, and f (x) = (sin
3 
x)(cos x) gives you
f
x
x dx
x
x dx
avg =
−
(
)
=
(
)
∫
∫
1
2
0
2
3
0
2
3
0
2
π
π
π
π
sin
(cos )
sin
(cos )
Next, use the substitution u = sin x so that du = cos x
 
dx. Find the new limits
of integration by noting that if x = π
2
, then u =
=
sin π
2
1, and if x = 0, then
u = sin 0 = 0. With these new values, you get the following:
2
2
2
4
2 1
4
0
1
2
3
0
2
3
0
1
4
0
1
π
π
π
π
π
π
sin
(cos )
x
x dx
u du
u
(
)
=
=






=
−
( )
=
∫
∫
739.
26
9
Using the average value formula with a = 0, b = 2, and f x
x
x
( ) =
+
2
3
1
gives you
f
x
x dx
avg = −
+
∫
1
2 0
1
2
3
0
2
Begin by using the substitution u = 1 + x
3
so that du = 3x
2 
dx, or 1
3
2
du x dx
=
. You can
also find the new limits of integration by noting that if x = 2, then u = 1 + 2
3
= 9, and
if x = 0, then u = 1 + 0
3
= 1. With these values, you have the following:
1
2 0
1
1
2
1
3
1
6
1
6
2
3
1
9
9
2
3
0
2
1
9
1 2
1
9
3 2
1
9
3
−
+
=
=
=






=
∫
∫
∫
x
x dx
u du
u du
u
2 2
32
1
1
9
27 1
26
9
−
(
)
=
−
=
(
)