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10
2004 Calculus #10
2004 Calculus #10
Source:
November 29, 2011
calculus
integration
trigonometry
induction
Problem Statement
Let
P
(
x
)
=
x
3
−
3
2
x
2
+
x
+
1
4
P(x)=x^3-\tfrac{3}{2}x^2+x+\tfrac{1}{4}
P
(
x
)
=
x
3
−
2
3
x
2
+
x
+
4
1
. Let
P
[
1
]
(
x
)
=
P
(
x
)
P^{[1]}(x)=P(x)
P
[
1
]
(
x
)
=
P
(
x
)
, and for
n
≥
1
n\ge1
n
≥
1
, let
P
n
+
1
(
x
)
=
P
[
n
]
(
P
(
x
)
)
P^{n+1}(x)=P^{[n]}(P(x))
P
n
+
1
(
x
)
=
P
[
n
]
(
P
(
x
))
. Evaluate:
∫
0
1
P
[
2004
]
(
x
)
d
x
.
\displaystyle\int_{0}^{1} P^{[2004]} (x) \ \mathrm{d}x.
∫
0
1
P
[
2004
]
(
x
)
d
x
.
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