Answers
Answered by
0
Answer:
To begin we must take:
5
x
+
12
x
2
+
5
x
+
6
and decompose it into its partial fractions. First factorise the denominator and then split the fraction up as follows:
5
x
+
12
(
x
+
3
)
(
x
+
2
)
=
A
x
+
3
+
B
x
+
2
Now, if we multiply the whole thing through by
(
x
+
3
)
(
x
+
2
)
then we should get an equation that will allow us to solve for
A
and
B
.
→
5
x
+
12
=
A
(
x
+
2
)
+
B
(
x
+
3
)
Now, to find
A
set
x
=
3
to cancel the second term and we get:
5
(
−
3
)
+
12
=
A
(
−
3
+
2
)
+
B
(
−
3
+
3
)
−
3
=
−
A
→
A
=
3
Now set
x
=
−
2
to obtain the value for for
B
.
→
5
(
−
2
)
+
12
=
A
(
−
2
+
2
)
+
B
(
−
2
+
3
)
→
B
=
2
So now we have that
A
=
3
and
B
=
2
we can re write the fraction given in the question as:
5
x
+
12
x
2
+
5
x
+
6
=
3
x
+
3
+
2
x
+
2
So we can now integrate:
∫
3
x
+
3
+
2
x
+
2
d
x
=
3
ln
(
x
+
3
)
+
2
ln
(
x
+
2
)
+
C
Similar questions