Answers and Explanations 565
Answers
901–1,001
956.
8
4
1
2
ln
ln
x
x
C
−
+ +
The given expression is
8
3
x
x
dx
+
∫
First factor the denominator of the integrand:
8
8
1
3
2
x
x x x
+
=
+
(
)
And then perform the fraction decomposition:
8
1
1
2
2
x x
A
x
Bx C
x
+
(
)
= +
+
+
Multiply both sides by x(x
2
+ 1) and simplify to get
x x
x x
x x
A
x
Bx C
x
A x
2
2
2
2
2
1
8
1
1
1
8
1
+
(
)
+
(
)








=
+
(
) + +
+






=
+
(
) + (
(
) ( )
B x C x
+
Expanding the right side, rearranging terms, and rewriting the left side yields the
following:
8
8
0
0
8
2
2
2
2
2
=
+ +
+
=
+
+
+
+
+ =
+
+
+
Ax
A Bx Cx
A B x Cx A
x
x
A B x Cx A
(
)
(
)
Equating coefficients gives you 0 = A + B, 0 = C, and 8 = A. Using 8 = A and 0 = A + B
gives you 0 = 8 + B, or –8 = B. These values give you the following integral:
8
1
8
8
1
2
3
2
x
x
dx
A
x
B x C
x
dx
x
x
x
dx
+
=
+
+
+






=
+
−
+






∫
∫
∫
To evaluate the first term in the integrand, use elementary antiderivatives:
8
8
x
dx
x C
=
+
∫
ln
. To evaluate the second term in the integrand, use a substitution on
−
+
∫
8
1
2
x
x
dx where u = x
2
+ 1 so that du = 2x
 
dx. Multiplying both sides of last equation by
–4 gives you –4
 
du = –8x
 
dx. Using these substitutions gives you the following:
−
+
= −
= −
+
= −
+ +
∫
∫
8
1
4
4
4
1
2
2
x
x
dx
u
du
u C
x
C
ln
ln
Combining the two antiderivatives gives you the solution:
8
8
1
8
4
1
2
2
x
x
x
dx
x
x
C
+ −
+






=
−
+ +
∫
ln
ln