Compute int_0^(oo) (sin(x))/(xe^x)dx

lunja55 2022-09-24 Answered
Compute 0 sin ( x ) x e x   d x
You can still ask an expert for help

Want to know more about Laplace transform?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

Guadalupe Reid
Answered 2022-09-25 Author has 8 answers
0 sin x x e x d x = 0 sin x 1 e s x d s d x = 1 0 sin ( x ) e s x d x d s = 1 L [ sin ( x ) ] ( s ) d s = 1 d s s 2 + 1 = π 2 tan 1 1 = π 4
Did you like this example?
Subscribe for all access
Jase Rocha
Answered 2022-09-26 Author has 2 answers
Let
I ( a ) = 0 sin ( x ) x e a x d x
where your integral is I(1). I′(a) is then:
I ( a ) = 0 sin ( x ) e a x d x
which is the Laplace Transform of the sine function. Thus
I ( a ) = 1 a 2 + 1
Integrating back we get:
I ( a ) = arctan ( a ) + C
To find the constant, we notice that
I ( 0 ) = 0 sin ( x ) x d x = π 2
So we get that C = π 2 and that
I ( 1 ) = arctan ( 1 ) + π 2 = π 4
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more