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Mathematics
Mathematics
Problem
If
I
n
=
∫
x
n
a
2
−
x
2
d
x
I_n = \int x^n \sqrt{a^2 - x^2} dx
I
n
=
∫
x
n
a
2
−
x
2
d
x
, prove that
I
n
=
−
x
n
−
1
(
a
2
−
x
2
)
3
/
2
(
n
+
2
)
+
(
n
−
1
)
(
n
+
2
)
a
2
I
n
−
2
.
I_n = -\frac{x^{n-1} (a^2 - x^2)^{3/2}}{(n+2)} + \frac{(n-1)}{(n+2)} a^2 I_{n-2}.
I
n
=
−
(
n
+
2
)
x
n
−
1
(
a
2
−
x
2
)
3/2
+
(
n
+
2
)
(
n
−
1
)
a
2
I
n
−
2
.
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