Integral of cos(ln(x)) dx: A Third Method Via Euler's Identity

\begin{align*}& \int \cos \left( \ln (x) \right) \: dx = \dfrac{x}{2} \left( \cos \left( \ln (x) \right) + \sin \left( \ln (x) \right) \right) + C \\& \\&{\rm{Proof}}: \\& \\& a= \ln (x) \Longrightarrow \int \cos (a) e^a \: da = \Re \left[ \int e^{ia} e^a \: da \right] \\& \\&\therefore \Re \left[ \dfrac{ e^{a(1+i)} }{1+i} \right] + C \Longrightarrow
\Re \left[ \left( \dfrac{1-i}{2} \right) e^{a(1+i)} \right] + C
\\& \\& = \left( \dfrac{e^a}{2} \right) \left( \cos a + \sin a \right) + C \\& \\&= \dfrac{x}{2} \left( \cos \left( \ln (x) \right) + \sin \left( \ln (x) \right) \right) + C
\end{align*}

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