I have t being the number of days since tickets went on sale and the function for daily sales is √(3t) + (99t)/(√(50+16t^2)) tickets/days. Find the total number of tickets sold during the week tickets were on sale. I'm pretty sure that given the specific days wants, the limits would be [7,14]

why [7,14]? Why not [0,7]?

In any case,
∫√(3t) + (99t)/(√(50+16t^2)) dt = 2/√3 t^(3/2) + 99/16 √(16t^2+50)