What is the closed-form of the following integral I=int_0^(oo) (e^(-x)cos(x))/(x^2+1)dx

steveo963200054

steveo963200054

Answered question

2022-09-10

What is the closed-form of the following integral
I = 0 e x cos ( x ) x 2 + 1 d x
If we replaced 1 x 2 + 1 by its integral representation, 0 e x t sin ( t ) d t , , we get that
I = 0 e x cos ( x ) ( 0 e x t sin ( t ) d t ) d x
{  reverse the order of integration  }
I = 0 sin ( t ) ( 0 e x ( t + 1 ) cos ( x ) d x ) d t
I = 0 ( t + 1 ) sin ( t ) ( t + 1 ) 2 + 1 d t
{  make the change of variable  t + 1 = u }
= 1 u sin ( u 1 ) u 2 + 1 d t
= cos ( 1 ) 1 u sin ( u ) u 2 + 1 d t sin ( 1 ) 1 u cos ( u ) u 2 + 1 d t
Have anyone an idea to finish the remaining integrals ?

Answer & Explanation

Willie Smith

Willie Smith

Beginner2022-09-11Added 18 answers

The Laplace transform of 1 1 + x 2 is well known
0 + e s x 1 + x 2 d x = C i ( s ) sin ( s ) + π cos ( s ) 2 S i ( s ) cos ( s )
where Ci(s),Si(s) is the cosine and sine integral, then note that
0 + e x cos ( x ) 1 + x 2 d x = R e ( 0 + e x + i x 1 + x 2 d x )
= R e ( C i ( 1 i ) sin ( 1 i ) + π cos ( 1 i ) 2 S i ( 1 i ) cos ( 1 i ) ) 0.47941
and I don't think we can be more explicit than that.

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