312
5.5. OTHER OPTIONS FOR FINDING ALGEBRAIC ANTIDERIVATIVES
This equivalent integral expression is straightforward to evaluate, and hence we find that
5x
x 2 − x − 2
dx =
10
3
ln |x − 2| +
5
3
ln |x + 1| + C.
It turns out that for any rational function R(x) =
P(x)
Q(x) where the degree of the polynomial
P is less than 7 the degree of the polynomial Q, the method of partial fractions can be
used to rewrite the rational function as a sum of simpler rational functions of one of the
following forms:
A
x − c
,
A
(x − c) n , or
Ax + B
x 2 + k
where A, B, and c are real numbers, and k is a positive real number. Because each of these
basic forms is one we can antidifferentiate, partial fractions enables us to antidifferentiate
any rational function.
A computer algebra system such as Maple, Mathematica, or WolframAlpha can be used to
find the partial fraction decomposition of any rational function. In WolframAlpha, entering
partial fraction 5x/(xˆ2-x-2)
results in the output
5x
x 2 − x − 2
=
10
3(x − 2)
+
5
3(x + 1)
.
We will primarily use technology to generate partial fraction decompositions of rational
functions, and then work from there to evaluate the integrals of interest using established
methods.
Activity 5.13.
For each of the following problems, evaluate the integral by using the partial fraction
decomposition provided.
(a)
1
x 2 − 2x − 3
dx,
given that
1
x 2 −2x−3
=
1/4
x−3 −
1/4
x+1
(b)
x 2 + 1
x 3 − x 2 dx,
given that
x 2 +1
x 3 −x 2 = −
1
x −
1
x 2 +
2
x−1
(c)
x − 2
x 4 + x 2 dx,
given that
x−2
x 4 +x 2 =
1
x −
2
x 2 +
−x+2
1+x 2
⊳
7 If the degree of P is greater than or equal to the degree of Q, long division may be used to write R as the
sum of a polynomial plus a rational function where the numerator’s degree is less than the denominator’s.
5.5. OTHER OPTIONS FOR FINDING ALGEBRAIC ANTIDERIVATIVES
This equivalent integral expression is straightforward to evaluate, and hence we find that
5x
x 2 − x − 2
dx =
10
3
ln |x − 2| +
5
3
ln |x + 1| + C.
It turns out that for any rational function R(x) =
P(x)
Q(x) where the degree of the polynomial
P is less than 7 the degree of the polynomial Q, the method of partial fractions can be
used to rewrite the rational function as a sum of simpler rational functions of one of the
following forms:
A
x − c
,
A
(x − c) n , or
Ax + B
x 2 + k
where A, B, and c are real numbers, and k is a positive real number. Because each of these
basic forms is one we can antidifferentiate, partial fractions enables us to antidifferentiate
any rational function.
A computer algebra system such as Maple, Mathematica, or WolframAlpha can be used to
find the partial fraction decomposition of any rational function. In WolframAlpha, entering
partial fraction 5x/(xˆ2-x-2)
results in the output
5x
x 2 − x − 2
=
10
3(x − 2)
+
5
3(x + 1)
.
We will primarily use technology to generate partial fraction decompositions of rational
functions, and then work from there to evaluate the integrals of interest using established
methods.
Activity 5.13.
For each of the following problems, evaluate the integral by using the partial fraction
decomposition provided.
(a)
1
x 2 − 2x − 3
dx,
given that
1
x 2 −2x−3
=
1/4
x−3 −
1/4
x+1
(b)
x 2 + 1
x 3 − x 2 dx,
given that
x 2 +1
x 3 −x 2 = −
1
x −
1
x 2 +
2
x−1
(c)
x − 2
x 4 + x 2 dx,
given that
x−2
x 4 +x 2 =
1
x −
2
x 2 +
−x+2
1+x 2
⊳
7 If the degree of P is greater than or equal to the degree of Q, long division may be used to write R as the
sum of a polynomial plus a rational function where the numerator’s degree is less than the denominator’s.