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Question

# Evaluate f(3). $f ( x ) = \int _ { 0 } ^ { \infty } x ^ { - t } d t$

Solution

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$\textbf{To Solve :}$

$\int_{0}^{\infty} 3^{-t} \,dt$

$\textbf{Solution :}$

\begin{align*} \int_{0}^{\infty} 3^{-t} \,dt &= \lim\limits_{b \to \infty} \int_{0}^{b} 3^{-t} \,dt \\ &= \lim\limits_{b \to \infty} \eval{\dfrac{3^{-t}}{-\ln 3}}_{0}^{b} \\ &= \lim\limits_{b \to \infty} \eval{\dfrac{3^{-t}}{\ln 3}}_{b}^{0} \\ &= \lim\limits_{b \to \infty} \qty(\dfrac{3^{-0}}{\ln 3} - \dfrac{3^{-b}}{\ln 3}) \\ &= \dfrac{1}{\ln 3} - \dfrac{3^{-\infty}}{\ln 3} \\ &= \dfrac{1}{\ln 3} - \dfrac{0}{\ln 3} \\ &= \dfrac{1}{\ln 3} \end{align*}

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