So we have
Make the substitution to get
The integral is now in the form of a gamma function
i know what the laplace transform of t^(3/2) is from looking at a table but how do you prove it? i applied the definition of the laplace transform and ended up doing integration by parts 2 times so that the integral i ended up with has e^(-st)*t^(-1/2). when i expanded again using integration by parts, the (-1/s)e^(-st)*t^(-1/2) evaluates to negative infinite which means it diverges. so i can't proceed further. how do you derive the laplace transform of t^(3/2)?