Evaluate int_0^(oo) 2u sin u e^(-su^2) du

lemondedeninaug

lemondedeninaug

Answered question

2022-09-09

Evaluate int_0^(oo) 2u sin u e^(-su^2) du
This integral emerged while finding Laplace Transform of sin ( u )
As a start, I used:
I = 2 u = 0 v = 0 u cos v   d v d u = 1 s 0 cos v e s v 2 d v
Again here I am stuck.

Answer & Explanation

Karson French

Karson French

Beginner2022-09-10Added 15 answers

(We assume Re s > 0)
I = 0 2 u sin u e s u 2 d u = Im u e i u s u 2 d u = Im d i d α e i α u s u 2 d u | α = 1 .
Now, we have
e i α u s u 2 d u = e α 2 / 4 s e s ( u i α / 2 s ) 2 d u = π s e α 2 / 4 s .
As a result, we obtain
I = π 2 s 3 / 2 e 1 / 4 s .

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