Question

Evaluate :${\int }_0^{\infty }{x}^7{e}^{\frac{x}{2}}dx$

Answer 1

${\int }_0^{\infty }{x}^7{e}^{\frac{x}{2}}dx$

Let $y=\frac{x}{2}\Rightarrow dy=\frac{1}{2}dx$

$={\int }_0^{\infty }{\left(2y\right)}^7{e}^{-y}2.dy$

$=2^8{\int }_0^{\infty }{y}^7{e}^{-y}.dy$

$=2^{8}7!$