Answers and Explanations 573
Answers
901–1,001
−
+
+ +
(
)
= −
+ +
+ +
=
+
−
+ +
=
+ +
∫
1
1
2
2
1 2
2
2
1
4 4
2
2
u
u
du
u
u
C
u
u
C
u
u
ln
ln
ln (
)
ln
ln
u u
C
e
e
e
C
x
x
x
+
+
=
+
+
+
+
1
4
4
1
2
ln
964.
convergent, 1
2
Begin by rewriting the integral using a limit:
1
1
1
1
2
1
2
1
(
)
lim (
)
x
dx
x
dx
a
a
+
=
+
∞
→∞
∫
∫
First evaluate
1
1
2
(
)
x
dx
+
∫
using the substitution u = x + 1 so that du = dx. Using these
substitutions gives you
1
1
1
2
2
1
u
du
u du
u
C
x
C
∫
∫
=
= −
+
= − +
+
−
−
Because
1
1
1
1
2
(
)
x
dx
x
C
+
= − +
+
∫
, you have the following after replacing the limit at
infinity and the limits of integration:
lim (
)
lim
a
a
a
a
x
dx
x
→∞
→∞
+
=
− +






∫
1
1
1
1
2
1
1
Now evaluate the limit:
lim
lim
a
a
a
x
a
→∞
→∞
− +






=
− +
+ +
(
)
= +
( )
=
1
1
1
1
1
1 1
0 1
2
1
2
1
Because the value of the integral is finite, the improper integral is convergent.