Express the following as a proper rational fraction and find the integral
of the function (a)
please help me out
explain in details
Answers
Step-by-step explanation:
Recall that a rational function is a ratio of two polynomials
P
(
x
)
Q
(
x
)
.
We will assume that we have a proper rational function in which the degree of the numerator is less than the degree of the denominator.
In order to convert improper rational function into a proper one, we can use long division:
P
(
x
)
Q
(
x
)
=
F
(
x
)
+
R
(
x
)
Q
(
x
)
,
where
F
(
x
)
is a polynomial,
R
(
x
)
Q
(
x
)
is a proper rational function.
To integrate a proper rational function, we can apply the method of partial fractions.
This method allows to turn the integral of a complicated rational function into the sum of integrals of simpler functions.
The denominators of the partial fractions can contain nonrepeated linear factors, repeated linear factors, nonrepeated irreducible quadratic factors, and repeated irreducible quadratic factors.
To evaluate integrals of partial fractions with linear or quadratic denominators, we use the following
6
formulas:
1.
∫
A
d
x
a
x
+
b
=
A
ln
|
a
x
+
b
|
2.
∫
A
d
x
(
a
x
+
b
)
k
=
A
a
(
1
−
k
)
(
a
x
+
b
)
k
−
1
For partial fractions with irreducible quadratic denominators, we first complete the square:
a
x
2
+
b
x
+
c
=
a
[
(
x
+
b
2
a
)
2
+
4
a
c
−
b
2
4
a
2
]
.
Hence, we can write:
∫
A
x
+
B
(
a
x
2
+
b
x
+
c
)
k
d
x
=
∫
A
t
+
B
′
[
a
(
t
2
+
m
2
)
]
k
d
t
=
1
a
k
∫
A
t
+
B
′
(
t
2
+
m
2
)
k
d
t
,
where
t
=
x
+
b
2
a
,
m
2
=
4
a
c
−
b
2
4
a
2
,
B
′
=
B
−
A
b
2
a
.
The possible cases for fractions with quadratic denominators are covered by the following integrals:
3.
∫
t
d
t
t
2
+
m
2
=
1
2
ln
(
t
2
+
m
2
)
4.
∫
d
t
t
2
+
m
2
=
1
m
arctan
t
m
5.
∫
t
d
t
(
t
2
+
m
2
)
k
=
1
2
(
1
−
k
)
(
t
2
+
m
2
)
k
−
1
Finally, the integral
∫
d
t
(
t
2
+
m
2
)
k
can be evaluated in
k
steps using the reduction formula