By parts over an infinite tail

MediumCalculus~4m

Evaluate $\displaystyle\int_0^{\infty} x\,e^{-2x}\,dx$ . Give a decimal to four places.

Show hints (2)+

By parts with $u=x$ , $dv=e^{-2x}dx$ , then take the limit as the bound $\to\infty$ .
The boundary term $\tfrac{b}{2}e^{-2b}\to 0$ ; answer is $\tfrac1{a^2}$ with $a=2$ .

Answer

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