Using Laplace Transforms to solve int_0^(oo) sin(x^2)dx

Altenbraknz 2022-09-20 Answered
Using Laplace Transforms to solve 0 sin ( x 2 ) d x
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Answers (1)

baselulaox
Answered 2022-09-21 Author has 8 answers
You can first substitute u = x 2 , use a useful property of the Laplace Transform and the Beta function and then utilize the reflection formula for the Gamma function
0 sin ( x 2 ) d x = 1 2 0 sin u u d u = 1 2 π 0 d s s ( s 2 + 1 ) = 1 2 π 0 1 ν 1 / 4 ( 1 + ν ) d ν 2 ν = 1 2 π 0 d ν ν 3 / 4 ( 1 + ν ) = 1 2 π B ( 1 / 4 , 3 / 4 ) = 1 2 π Γ ( 1 / 4 ) Γ ( 3 / 4 ) Γ ( 1 ) = 1 2 π π 2 = π 8
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