Question Number 101959 by MathQoutes last updated on 05/Jul/20

    Starting from          y=(4/(√π))t^((3/2) ) ∫_0 ^∞ x^3 e^(−tx^2 ) dx  find   (π/8)=?

Answered by mathmax by abdo last updated on 06/Jul/20

the question is not clear  let find y(t) =(4/(√π))t^(3/2)  ∫_0 ^∞  x^3  e^(−tx^2 )  dx for t>0  we do the changement x(√t)=u ⇒  y(t) =(4/(√π))t^(3/2)  ∫_0 ^∞  ((u/(√t)))^(3 )  e^(−u^2 )  (du/(√t))  =(4/(√(πt))) ∫_0 ^∞   u^3  e^(−u^2 ) du  =_(u = (√z))   (4/(√(πt))) ∫_0 ^∞  z^(3/2)  e^(−z)  (dz/(2(√z)))  =(2/(√(πt)))∫_0 ^∞  z e^(−z)  dz  we know  Γ(z) =∫_0 ^∞  z^(x−1)  e^(−z)  dt  ⇒∫_0 ^∞  z e^(−z ) dz =Γ(2) =1!=1  ⇒y(t) =(2/(√(πt)))