Question

Find the value of ${\int }_0^{\infty }x{e}^{-x}dx$

• A. 1
• B. -1
• C. 0
• D. None of these

Answer 1

The correct option is A

1

${\int }_0^{\infty }x{e}^{-x}dx={lim}_{b\to \infty }{\int }_0^b{e}^{-x}xdx={lim}_{b\to \infty }{(-x{e}^{-x}-e^{-x})}_0^b={lim}_{b\to \infty }-b{e}^{-b}-e^{-b}+1={lim}_{b\to \infty }\frac{-(b+1)}{e^b}+{lim}_{b\to \infty }1={lim}_{b\to \infty }\frac{-1}{e^b}+{lim}_{b\to \infty }1\text{using L'Hospital's theorem}=1$