Answers and Explanations 561
Answers
901–1,001
950.
x
x
C
+
− +
5
5
ln
The given integral is
x
x
dx
−
∫ 5
Notice that the degree of the numerator is equal to the degree of the denominator, so
you must divide. You could use long division, but for simple expressions of this type,
you can simply subtract 5 and add 5 to the numerator and then split the fraction:
x
x
dx
x
x
dx
x
x
x
dx
x
dx
−
=
− +
−
=
−
−
+ −
(
)
=
+ −
(
)
∫
∫
∫
∫
5
5 5
5
5
5
5
5
1
5
5
(
)
Now apply elementary antiderivative formulas to get the solution:
1
5
5
5
5
+ −
(
) = + − +
∫ x
dx x
x
C
ln
951.
x
x
x
C
2
2
6
36
6
−
+
+ +
ln
The given integral is
x
x
dx
2
6
+
∫
The degree of the numerator is greater than or equal to the degree of the denominator,
so use polynomial long division to get the following:
x
x
dx
x
x
dx
2
6
6
36
6
+
=
− + +
(
)
∫
∫
Then apply basic antiderivative formulas:
x
x
dx x
x
x
C
− + +
(
) = − +
+ +
∫
6
36
6
2
6
36
6
2
ln
952.
7
9
4 2
9
5
ln
ln
x
x
C
+ +
− +
The given integral is
x
x
x
dx
−
+
−
∫
3
4
5
(
)(
)
First perform a fraction decomposition on the integrand:
x
x
x
A
x
B
x
−
+
−
= +
+ −
3
4
5
4
5
(
)(
)