please solve this question and explain it
Answers
answer
∫
x
4
x
2
−
1
d
x
=
1
3
x
3
+
x
+
1
2
ln
∣
∣
∣
x
−
1
x
+
1
∣
∣
∣
+
c
Explanation:
For the integrand
x
4
x
2
−
1
, we perform a long division to get it into an integrable form.
x
4
x
2
−
1
=
x
4
−
1
+
1
x
2
−
1
=
(
x
2
−
1
)
(
x
2
+
1
)
x
2
−
1
+
1
x
2
−
1
=
=
x
2
+
1
+
1
(
x
+
1
)
(
x
−
1
)
We now perform a partial fraction decomposition on the 2nd term
1
(
x
+
1
)
(
x
−
1
)
=
(
x
+
1
)
−
(
x
−
1
)
2
(
x
+
1
)
(
x
−
1
)
=
x
+
1
2
(
x
+
1
)
(
x
−
1
)
−
x
−
1
2
(
x
+
1
)
(
x
−
1
)
=
1
2
(
x
−
1
)
−
1
2
(
x
+
1
)
So
x
4
x
2
−
1
=
x
2
+
1
+
1
2
(
x
−
1
)
−
1
2
(
x
+
1
)
and we can now integrate:
∫
x
4
x
2
−
1
d
x
=
∫
x
2
+
1
+
1
2
(
x
−
1
)
−
1
2
(
x
+
1
)
d
x
=
1
3
x
3
+
x
+
1
2
ln
|
x
−
1
|
−
1
2
ln
|
x
+
1
|
+
c
=
1
3
x
3
+
x
+
1
2
ln
∣
∣
∣
x
−
1
x +1
∣
∣
∣ +c
Step-by-step explanation:
The spectrum of light emitted by a glowing solid is
A) Continuous spectrum B) Line spectrum C) Band spectrum D) Absorption spectrum