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2014 BMT Spring
20
2014 BMT Team 20 integral
2014 BMT Team 20 integral
Source:
January 6, 2022
algebra
calculus
integration
Problem Statement
A certain type of Bessel function has the form
I
(
x
)
=
1
π
∫
0
π
e
x
cos
θ
d
θ
I(x) = \frac{1}{\pi} \int_0^{\pi}e^{x \cos \theta} d\theta
I
(
x
)
=
π
1
∫
0
π
e
x
c
o
s
θ
d
θ
for all real
x
x
x
. Evaluate
∫
0
∞
x
I
(
2
x
)
e
−
x
2
d
x
\int_0^{\infty} x I(2x) e^{-x^2}dx
∫
0
∞
x
I
(
2
x
)
e
−
x
2
d
x
.
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