Answers and Explanations 575
Answers
901–1,001
968.
convergent, 32
3
Notice that the function f (x) = (x – 1)
−1/4
has an infinite discontinuity when x = 1, which
is included in the interval of integration, so you have an improper integral. Begin by
writing the integral using a limit:
(
)
lim (
)
x
d x
x
dx
a
a
−
=
−
−
→
−
∫
∫
+
1
1
1 4
1
17
1
1 4
17
Notice that (
)
(
)
/
x
d x
x
C
−
=
−
+
−
∫ 1
4
1
3
1 4
3 4
so that
lim (
)
lim
(
)
a
a
a
a
x
d x
x
→
−
→
+
+
−
=
−






∫
1
1 4
17
1
3 4
17
1
4
1
3
Substituting in the limits of integration and evaluating the limit gives you
lim
lim
a
a
a
x
a
→
→
+
+
−
(
)








=
−
(
) −
−
(
)



1
3 4
17
1
3 4
3 4
4
1
3
4 17 1
3
4
1
3
 




= 32
3
The answer is finite, so the integral is convergent.
969.
divergent
Begin by rewriting the integral using a limit:
1
1
1
1
3
3
x
dx
x
dx
a
a
+
=
+
−∞
−
→−∞
−
∫
∫
lim
To evaluate this integral, use elementary antiderivatives:
lim
lim ln
lim ln
ln
a
a
a
a
a
x
dx
x
a
→−∞
−
→−∞
−
→−∞
+
=
+
(
)
=
− + −
+
(
)
∫
1
1
1
3 1
1
3
3
Because lim ln
a
a
→−∞
+ = ∞
1
, the integral is divergent.