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Today's Calculation Of Integral
2012 Today's Calculation Of Integral
831
831
Part of
2012 Today's Calculation Of Integral
Problems
(1)
Today's calculation of Integral 831
Source: 1977 Tsukuba Univrersity entrance exam
7/3/2012
Let
n
n
n
be a positive integer. Answer the following questions.(1) Find the maximum value of
f
n
(
x
)
=
x
n
e
−
x
f_n(x)=x^{n}e^{-x}
f
n
(
x
)
=
x
n
e
−
x
for
x
≥
0
x\geq 0
x
≥
0
.(2) Show that
lim
x
→
∞
f
n
(
x
)
=
0
\lim_{x\to\infty} f_n(x)=0
lim
x
→
∞
f
n
(
x
)
=
0
.(3) Let
I
n
=
∫
0
x
f
n
(
t
)
d
t
I_n=\int_0^x f_n(t)\ dt
I
n
=
∫
0
x
f
n
(
t
)
d
t
. Find
lim
x
→
∞
I
n
(
x
)
\lim_{x\to\infty} I_n(x)
lim
x
→
∞
I
n
(
x
)
.
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