$Put, \\ \hspace{1cm} y^2=t \,\,\,\,=\gt \,\,\,\,y=t^{1/2}\,\,\,\, =\gt\,\,\,\,dy=\frac{1}{2}t^{-1/2}dt \\ \hspace{3cm} $

$ \\ \int_{0}^{\infty} t^{1/4}e^{-t}\frac{1}{2}t^{1/2}dt = \int_{0}^{\infty}e^{-t} t^{-1/4}\frac{1}{2}t^{1/2}dt \\ \hspace{6cm} $

$ \frac{1}{2} \int_{0}^{\infty}e^{-t} t^{-1/4} dt \,=\, \int_{0}^{\infty}e^{-t} t^{-3/4}dt \\ \hspace{6cm} $

$ \frac{1}{2} \sqrt{3/4} * \frac{1}{2} \sqrt{1/4} = \frac{1}{4} \sqrt\frac{3}{4} \sqrt\frac{1}{4} $
$= \frac{1}{4}\sqrt2 \,\pi = \frac{\pi}{2\sqrt{2}} \\ $

$ \hspace{9cm}$

$(as \sqrt\frac{3}{4}\,\sqrt\frac{1}{4}= \frac{\pi}{\sin(\frac{\pi}{4})}=\frac{\pi}{1/\sqrt{2}}=\sqrt{2}\,\pi )$