Answers and Explanations 583
Answers
901–1,001
To find
x
x
dx
2
4
+
∫
, let u = x
2
+ 4 so that du = 2x
 
dx, or 1
2
du x dx
=
. This substitution
gives you
x
x
dx
u
du
u C
x
C
2
2
4
1
2
1
1
2
1
2
4
+
=
=
+
=
+ +
∫
∫
ln
ln
Because
x
x
dx
x
C
2
2
4
1
2
4
+
=
+ +
∫
ln
, you have
lim
lim ln
a
a
a
a
x
x
dx
x
→∞
→∞
+
=
+
∫ 2
0
2
0
4
1
2
4
Evaluating the limit gives you
lim ln
lim ln
ln
a
a
a
x
a
→∞
→∞
+
=
+ −
(
) = ∞
1
2
4
1
2
4
4
2
0
2
Therefore, the integral is divergent.
980.
convergent, − ( )
3
2
7
2 3
Begin by rewriting the integral using a limit:
1
8
1
8
3
1
8
8
3
1
x
dx
x
dx
a
a
−
=
−
∫
∫
→
−
lim
To evaluate
1
8
3
x
dx
−
∫
, let u = x – 8 so that du = dx. This substitution gives you
1
2 3
3
2
8
3
1 3
2 3
2 3
u
du
u du
u
C
x
C
∫
∫
=
=
+
=
−
+
−
(
)
Because 1
3
2
8
3
2 3
u
du
x
C
∫
=
−
+
(
)
, you have
lim
lim
(
)
a
a
a
a
x
dx
x
→
→
−
−
−
=
−






∫
8
3
1
8
2 3
1
1
8
3
2
8
Evaluating the limit gives you
lim
lim
a
a
a
x
a
→
→
−
−
−
(
)






=
−
(
) −
−
(
)
(
)
8
2 3
1
8
2 3
2 3
3
2
8
3
2
8
3
2
1 8