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Today's Calculation Of Integral
2011 Today's Calculation Of Integral
762
Today's calculation of Integral 762
Today's calculation of Integral 762
Source: 1993 Kogakuin Univesity entrance exam/Engineering
November 5, 2011
calculus
integration
function
trigonometry
calculus computations
Problem Statement
Define a function
f
n
(
x
)
(
n
=
0
,
1
,
2
,
⋯
)
f_n(x)\ (n=0,\ 1,\ 2,\ \cdots)
f
n
(
x
)
(
n
=
0
,
1
,
2
,
⋯
)
by
f
0
(
x
)
=
sin
x
,
f
n
+
1
(
x
)
=
∫
0
π
2
f
n
′
(
t
)
sin
(
x
+
t
)
d
t
.
f_0(x)=\sin x,\ f_{n+1}(x)=\int_0^{\frac{\pi}{2}} f_n\prime (t)\sin (x+t)dt.
f
0
(
x
)
=
sin
x
,
f
n
+
1
(
x
)
=
∫
0
2
π
f
n
′
(
t
)
sin
(
x
+
t
)
d
t
.
(1) Let
f
n
(
x
)
=
a
n
sin
x
+
b
n
cos
x
.
f_n(x)=a_n\sin x+b_n\cos x.
f
n
(
x
)
=
a
n
sin
x
+
b
n
cos
x
.
Express
a
n
+
1
,
b
n
+
1
a_{n+1},\ b_{n+1}
a
n
+
1
,
b
n
+
1
in terms of
a
n
,
b
n
.
a_n,\ b_n.
a
n
,
b
n
.
(2) Find
∑
n
=
0
∞
f
n
(
π
4
)
.
\sum_{n=0}^{\infty} f_n\left(\frac{\pi}{4}\right).
∑
n
=
0
∞
f
n
(
4
π
)
.
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