Integrals: Royal Properties

\begin{align*}
& {\rm{Given}}:
\int_0^\infty f(x) dx \\& \\&(1) \: \rm{Queen \: Property} \\& \\&
\therefore u=(x)^{-1} \Longrightarrow
\int_0^\infty f[(x)^{-1}] (x)^{-2} dx \\& \\&(2) \: \rm{Jacksonian \: Property} \\& \\&
\therefore u=(x)^{-n} \Longrightarrow
n \int_0^\infty f[(x)^{-n}] (x)^{-(n+1)} dx \\& \\&(3) \: \rm{Prince \: Property} \\& \\&
\therefore u=\ln{(x+1)} \Longrightarrow
\int_0^\infty f[\ln{(x+1)}] (x+1)^{-1} dx \\& \\&(4) \: \rm{Joker \: Property} \\& \\&
\therefore u=e^{x^n}-1 \Longrightarrow
n \int_0^\infty f[e^{x^n}-1] (x)^{n-1}\mathrm{(e)}^{x^n} dx
\end{align*}

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