Question 1185369
It is given that {{{L(cos(7t)/sqrt(pi*t)) = e^(-7/s)/sqrt(s)}}}.


===> {{{int((cos(7t)/sqrt(pi*t))*e^(-st), dt, 0, infinity) = e^(-7/s)/sqrt(s)}}}, by applying the definition of the Laplace transform to the hypothesis.


===> {{{(d/ds)int((cos(7t)/sqrt(pi*t))*e^(-st), dt, 0, infinity) = (d/ds)(e^(-7/s)/sqrt(s))}}}, i.e., taking derivatives wrt s.


===> {{{int((cos(7t)/sqrt(pi*t))*e^(-st)*(-t), dt, 0, infinity) = -int((sqrt(t/pi))* cos(7t) *e^(-st), dt, 0, infinity)= (sqrt(s)*e^(-7/s)*(7/s^2) - e^(-7/s)*(1/(2sqrt(s))))/s = e^(-7/s)*(7/s^(5/2) - 1/(2s^(3/2)))}}}


===>  {{{int((sqrt(t/pi))* cos(7t) *e^(-st), dt, 0, infinity) =  e^(-7/s)*( 1/(2s^(3/2)) - 7/s^(5/2))}}}


Therefore, {{{L(sqrt(t/pi)* cos(7t)  ) =  e^(-7/s)*( 1/(2s^(3/2)) - 7/s^(5/2))}}}