Part II: The Answers
584
Answers
901–1,001
Noting that lim
a
a
→
−
−
(
) =
8
2 3
3
2
8
0, you get the following:
lim
a
a
→
−
−
(
) −
−
(
)
(
)
= −
−
(
)
= − −
( )
= − ( )
8
2 3
2 3
2 3
2 3
2
3
2
8
3
2
1 8
0 3
2
1 8
3
2
7
3
2
7
3 3
The answer is finite, so the integral is convergent.
981.
divergent
Begin by rewriting the integral using a limit:
tan
lim
tan
2
0
2
2
2
0
x dx
x dx
a
a
π
π
∫
∫
=
→ ( )
−
To evaluate tan
2 x dx
∫
, use a trigonometric identity:
tan
s ec
tan
2
2
1
x dx
x
dx
x x C
∫
∫
=
−
(
)
=
− +
Because tan
t an
2 x dx
x x C
∫
=
− + , you have
lim
tan
lim tan
a
a
a
a
x dx
x x
→ ( )
→ ( )
−
− ∫
=
−
(
)
π
π
2
2
0
2
0
Evaluating the limit gives you
lim tan
lim tan
t an
a
a
a
x x
a a
→ ( )
→ ( )
−
−
−
(
)
=
− −
−
(
)
(
)
π
π
2
0
2
0 0
However, because lim tan
a
a
→ ( )
−
= ∞
π
2
, you can conclude that
tan
2
0
2
x dx
π
∫
is divergent.
982.
divergent
Begin by rewriting the integral using a limit:
sin
lim sin
2
0
2
0
x dx
x dx
a
a
∞
→∞
∫
∫
=
To evaluate sin
2 x dx
∫
, use the trigonometric identity sin
c os( )
2
1
2
1
2
x
x
=
−
(
) so that
sin
c os( )
sin( )
2
1
2
1
2
1
2
1
2
2
x dx
x dx
x
x
C
∫
∫
=
−
(
)
=
−
(
) +