How do you integrate f(x)=x−2(x2−3)(x−3)(x−1) using partial fractions?
Answers
Answer:
∫
3
x
(
x
+
2
)
(
x
−
1
)
d
x
=
ln
(
A
(
x
+
2
)
2
|
x
−
1
|
)
Explanation:
Let us find the partial fraction decomposition of the integrand:
Let
3
x
(
x
+
2
)
(
x
−
1
)
≡
A
x
+
2
+
B
x
−
1
∴
3
x
(
x
+
2
)
(
x
−
1
)
=
A
(
x
−
1
)
+
B
(
x
+
2
)
(
x
+
2
)
(
x
−
1
)
∴
3
x
=
A
(
x
−
1
)
+
B
(
x
+
2
)
This is an identity and valid
∀
x
∈
R
Put
x
=
−
2
⇒
−
6
=
−
3
A
+
0
⇒
A
=
2
Put
x
=
1
⇒
3
=
0
+
B
(
3
)
⇒
B
=
1
So the partial fraction decomposition is:
3
x
(
x
+
2
)
(
x
−
1
)
≡
2
x
+
2
+
1
x
−
1
And so the integral can be written as:
∫
3
x
(
x
+
2
)
(
x
−
1
)
d
x
=
∫
2
x
+
2
+
1
x
−
1
d
x
∴
∫
3
x
(
x
+
2
)
(
x
−
1
)
d
x
=
2
∫
1
x
+
2
d
x
+
∫
1
x
−
1
d
x
∴
∫
3
x
(
x
+
2
)
(
x
−
1
)
d
x
=
2
ln
|
x
+
2
|
+
ln
|
x
−
1
|
+
ln
A
(
ln
A
=constant)
∴
∫
3
x
(
x
+
2
)
(
x
−
1
)
d
x
=
ln
(
A
(
x
+
2
)
2
|
x
−
1
|
)
Step-by-step explanation: