Right option is (b) \(\frac{15}{8} \frac{√π}{s^{7/2}}\)
The explanation: \(g(t)=t^{5/2}=\frac{5}{2} \int_0^t t^{\frac{3}{2}} dt=\frac{15}{4} \int_0^t \int_0^t √t dt dt\)
let f(t)=√t, hence, F(s)=\(\frac{\sqrt{π}}{2s^{\frac{3}{2}}}\)
hence, G(s)=\(\frac{15}{4} \,\frac{1}{s^2} \,F(s)=\frac{15}{8} \frac{√π}{s^{7/2}}\)