Question 1209758
.


Strictly saying, the problem and the question do not make sense,

since the integral is some certain real number - not a random variable.



So, my diagnosis is that this "problem" is a kind of nonsense.


Soup of words and mathematical symbols with no sense.



/////////////////////\\\\\\\\\\\\\\\\\\\\\



In a normal human language,  you want to find  X  in a way,  that 


for any  x > X  the integral of  e(-t^2)|sin(t)|  from  x  to infinity be less than  (1-0.96)/2 = 0.02.



                   A physicist wouldn't bother.


        He would say: just replace   |sin(t)|   by   1  (one unit)   and look for  X  such

        that for any  x > X  integral of   e(-t^2)   to infinity is less than  0.02.



            It is because   1   " majorizes "   |sin(x)|.



Then you come to the   STANDARD   normal distribution function.


For it,  you know from the  " empirical rule ",  that the  95%  symmetric confidence interval 
is two standard deviations from the mean,  which is zero in your case.


I think,  that for you it is the same now,  96%  confidence or  95% confidence.


So,  with this rude approach,  the value  X = 2   (two standard deviations)   is what you need.


It is how a physicist or an engineer would solve/answer your question 
after 5 minutes thinking and without making complicated calculations.


Or any other person, having common sense in his mind.



Let me know if it makes sense to you.