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Jozsef Wildt International Math Competition
2019 Jozsef Wildt International Math Competition
W. 10
Prove this identity on integration
Prove this identity on integration
Source: 2019 Jozsef Wildt International Math Competition-W. 10
May 18, 2020
integration
trigonometry
Exponents
calculus
Problem Statement
If
s
i
(
x
)
=
−
∫
x
∞
(
sin
t
t
)
d
t
;
x
>
0
{si}(x) =- \int \limits_{x}^{\infty}\left(\frac{\sin t}{t}\right)dt; x>0
s
i
(
x
)
=
−
x
∫
∞
(
t
s
i
n
t
)
d
t
;
x
>
0
then
∫
e
e
2
(
1
x
(
s
i
(
e
4
x
)
−
s
i
(
e
3
x
)
)
)
d
x
=
∫
3
e
4
(
1
x
(
si
(
e
2
x
)
−
s
i
(
e
x
)
)
)
d
x
\int \limits_{e}^{e^2} \left(\frac{1}{x}\left(si\left(e^4x\right)-si\left(e^3x\right)\right)\right)\,dx=\int \limits_{3}^{e^4} \left(\frac{1}{x}\left(\operatorname{si}\left(e^2x\right)-si\left(ex\right)\right)\right)dx
e
∫
e
2
(
x
1
(
s
i
(
e
4
x
)
−
s
i
(
e
3
x
)
)
)
d
x
=
3
∫
e
4
(
x
1
(
si
(
e
2
x
)
−
s
i
(
e
x
)
)
)
d
x
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