Starting-from-y-4-pi-t-3-2-0-x-3-e-tx-2-dx-find-pi-8-

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

S t a r t i n g f r o m

y = \frac{4}{\sqrt{π}} t^{\frac{3}{2}} \int_{0}^{\infty} x^{3} e^{- {t x}^{2}} d x

f i n d \frac{π}{8} = ?

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

the question is not clear let find y (t) = \frac{4}{\sqrt{π}} t^{\frac{3}{2}} \int_{0}^{\infty} x^{3} e^{- {tx}^{2}} dx for t > 0

we do the changement x \sqrt{t} = u \Rightarrow

y (t) = \frac{4}{\sqrt{π}} t^{\frac{3}{2}} \int_{0}^{\infty} {(\frac{u}{\sqrt{t}})}^{3} e^{- u^{2}} \frac{du}{\sqrt{t}}

= \frac{4}{\sqrt{π t}} \int_{0}^{\infty} u^{3} e^{- u^{2}} du =_{u = \sqrt{z}} \frac{4}{\sqrt{π t}} \int_{0}^{\infty} z^{\frac{3}{2}} e^{- z} \frac{dz}{2 \sqrt{z}}

= \frac{2}{\sqrt{π t}} \int_{0}^{\infty} z e^{- z} dz we know Γ (z) = \int_{0}^{\infty} z^{x - 1} e^{- z} dt \Rightarrow \int_{0}^{\infty} z e^{- z} dz = Γ (2) = 1! = 1

\Rightarrow y (t) = \frac{2}{\sqrt{π t}}