integral of this function from 0 to infinity : sinh(x^(0.5))*exp(-x)?

Question

i need analyctical soloution for this integral

Answer 1

(1/2) (√π ) e^(1/4)

METHOD 1

Use WIMS to evaluate the integral to 30 decimals. Use the Inverse Symbolic Calculator to get an answer.

METHOD 2

Using Wolfram Integrator, the integral is

F(x) =

(1/4) { -2 e^(- x -√x) [ -1 + e^(2√x) ] +
e^(1/4) √π [ erf( √x - 1/2 ) + erf( √x + 1/2 ) ] }

The answer is F(∞) - F(0):

F(∞) = (1/2) (√π ) e^(1/4)

using erf(∞) = 1

F(0) = 0

using erf( -x ) = - erf( x )

I am curious where this problem comes from.

Answer 2

use the chain rule and product rule for differentation,

sinh(x)' = cosh(x)
exp'(x) = exp( x)
and x^0.5 ' = 0.5 1/(x^0.5)

productrule (f*g) = f'g - g'f
chainrule f(g) ' = f'(g) * g'

with this info you should be abl;e to find the analitical solution
cheers.