Answers and Explanations 563
Answers
901–1,001
Applying elementary antiderivatives gives you the following:
−
+
+ −








= −
+ +
−
(
)
=
− −
∫
1 2
1
1 2
1
1
2
1 1
2
1
1
2
1
4
5
4
5
x
x
dx
x
x
x
x
ln
ln
ln
ln + +
(
)
=
−
+
=
−
+
−
−
+






=
−
1
1
2
1
1
1
2
5 1
5 1
4 1
4 1
1
2
2
3
3
5
4
5
4
5
ln
ln
ln
ln
ln
x
x
 





=
( )






= ( )
=





 =






1
2
10
9
10
9
10
9
10
3
1 2
ln
ln
ln
ln
954.
3
1
2
1
ln x
x
C
+ − +
+
The given integral is
3
5
2
1
2
x
x
x
dx
+
+
+
∫
Begin by factoring the denominator:
3
5
2 1
3
5
1
2
2
x
x
x
x
x
+
+
+
=
+
+
(
)
And then perform a fraction decomposition:
3
5
1
1
1
2
2
x
x
A
x
B
x
+
+
= +
+
+
(
)
(
)
Multiplying both sides by (x + 1)
2
gives you the following:
(
) (
)
(
)
(
)
(
)
x
x
x
x
A
x
B
x
x
A x
B
+
+
+
=
+
+
+
+






+ =
+ +
1 3
5
1
1
1
1
3
5
1
2
2
2
2
Expanding the right side and equating coefficients gives you 3x + 5 = Ax + (A + B) so
that A = 3 and A + B = 5; in turn, that gives you 3 + B = 5 so that B = 2. Now you have
3
5
2 1
1
1
3
1
2
1
2
2
2
x
x
x
dx
A
x
B
x
dx
x
x
dx
+
+
+
=
+
+
+






=
+
+
+






∫
∫
∫
(
)
(
)