Part II: The Answers
576
Answers
901–1,001
970.
divergent
Begin by rewriting the improper integral using a definite integral and a limit; then
integrate:
x
dx
x
dx
x
x
a
a
a
a
2
2
2
2
3
2
5
5
3
5
−
(
)
=
−
(
)
=
−


 


 
=
−∞
→−∞
→−∞
∫
∫
lim
lim
lim m
( )
a
a
a
→−∞
−
−
−












2
3
5 2
3
5
3
3
Notice that lim
lim
a
a
a
a
a
a
→−∞
→ −∞
−





 =
−






3
3
3
5
15
3
. Because the first term in the numerator is
the dominant term, you have
lim
lim
a
a
a
a
a
→−
→
∞
− ∞
−





 =





 = −∞
3
3
15
3
3
Therefore, it follows that
lim
( )
( )
a
a
a
→−∞
−
−
−











 =
−
+ ∞
2
3
5 2
3
5
2
3
5 2
3
3
3
and you can conclude the integral is divergent.
971.
divergent
Begin by splitting up the integral into two separate integrals, using limits to rewrite
each integral:
3
3
3
4
4
0
4
0
−
(
)
=
−
(
) +
−
(
)
−∞
∞
→−
→
∞
∞
∫
∫
∫
x dx
x dx
x dx
a
a
b
b
lim
lim
Evaluate the first integral by integrating; then evaluate the resulting limit:
lim
lim
lim ( )
a
a
a
a
a
x dx
x x
→−∞
→−∞
→−∞
−
(
)
=
−


 


 
=
−
∫ 3
3
5
3 0 0
5
4
0
5
0
5 − −
−












3
5
5
a a
Notice that lim
lim
a
a
a a
a a
→−∞
→ −∞
−





 =
−






3
5
15
5
5
5
. Because the second term in the numerator
is the dominant term, you have
lim
lim
a
a
a a
a
→−
→
∞
− ∞
−





 =
−





 = ∞
15
5
5
5
5