Answers and Explanations 579
Answers
901–1,001
And as b approaches ∞, you have
lim
lim
tan
lim tan
b
b
b
b
b
x
x
dx
x
b
→∞
→∞
−
→∞
−
+
=
( )






=
∫
2
6
0
1
3
0
1
1
1
3
1
3
3 3
1
1
3
0
1
3 2
0
6
( ) −
( ) )
(
= ( ) −
=
−
tan
π
π
Combining the two values gives you the answer:
lim
lim
a
a
b
b
x
x
dx
x
x
dx
→−
→
∞
∞
+
+
+
= +
=
=
∫
∫
2
6
0
2
6
0
1
1
6 6
2
6
3
π π
π
π
The answer is finite, so the integral is convergent.
Note: If you noticed that the integrand is even, you could’ve simply computed one
of the integrals and multiplied by 2 to arrive at the solution.
975.
convergent, 1
4
Begin by rewriting the integral using a limit:
xe dx
xe dx
x
a
x
a
−
∞
→∞
−
∫
∫
=
2
0
2
0
lim
To evaluate xe dx
x
−
∫
2
, use integration by parts. If u = x, then du = dx, and if dv = e
−2x
,
then v
e
x
= −
−
1
2
2 :
xe dx x
e
e
dx
xe
e
C
x
x
x
x
x
−
−
−
−
−
∫
∫
= −
(
) − −
(
)
= −
−
+
2
2
2
2
2
1
2
1
2
1
2
1
4
Because xe dx
xe
e
C
x
x
x
−
−
−
∫
= −
−
+
2
2
2
1
2
1
4
, it follows that
lim
lim
a
x
a
a
x
x
a
xe dx
xe
e
→∞
−
→∞
−
−
∫
=
−
−






2
0
2
2
0
1
2
1
4
Next, evaluate the limit:
lim
lim
a
x
x
a
a
a
a
xe
e
ae
e
→∞
−
−
→∞
−
−
−
−






=
−
−
− − ( )
1
2
1
4
1
2
1
4
1
2
0
2
2
0
2
2
e e
e
−
−
−
(
)






2 0
2 0
1
4
( )
( )