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Vojtěch Jarník IMC
2010 VJIMC
Problem 3
integral is bounded
integral is bounded
Source: VJIMC 2010 1.3
June 3, 2021
calculus
integration
Problem Statement
Prove that there exist positive constants
c
1
c_1
c
1
and
c
2
c_2
c
2
with the following properties: a) For all real
k
>
1
k>1
k
>
1
,
∣
∫
0
1
1
−
x
2
cos
(
k
x
)
d
x
∣
<
c
1
k
3
/
2
.
\left|\int^1_0\sqrt{1-x^2}\cos(kx)\text dx\right|<\frac{c_1}{k^{3/2}}.
∫
0
1
1
−
x
2
cos
(
k
x
)
d
x
<
k
3/2
c
1
.
b) For all real
k
>
1
k>1
k
>
1
,
∣
∫
0
1
1
−
x
2
sin
(
k
x
)
d
x
∣
<
c
2
k
.
\left|\int^1_0\sqrt{1-x^2}\sin(kx)\text dx\right|<\frac{c_2}k.
∫
0
1
1
−
x
2
sin
(
k
x
)
d
x
<
k
c
2
.
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