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Today's Calculation Of Integral
2012 Today's Calculation Of Integral
833
833
Part of
2012 Today's Calculation Of Integral
Problems
(1)
Today's calculation of Integral 833
Source: 2012 Kumamoto University entrance exam/medicine
7/4/2012
Let
f
(
x
)
=
∫
0
x
e
t
(
cos
t
+
sin
t
)
d
t
,
g
(
x
)
=
∫
0
x
e
t
(
cos
t
−
sin
t
)
d
t
.
f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.
f
(
x
)
=
∫
0
x
e
t
(
cos
t
+
sin
t
)
d
t
,
g
(
x
)
=
∫
0
x
e
t
(
cos
t
−
sin
t
)
d
t
.
For a real number
a
a
a
, find
∑
n
=
1
∞
e
2
a
{
f
(
n
)
(
a
)
}
2
+
{
g
(
n
)
(
a
)
}
2
.
\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.
∑
n
=
1
∞
{
f
(
n
)
(
a
)
}
2
+
{
g
(
n
)
(
a
)
}
2
e
2
a
.
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