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integral $\int \cos (\log_{e} x) d x$ equals (where $C$ is the constant of Integration)

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\frac{x}{2} [\sin (\log_{e} x) - \cos (\log_{c} x)] + C

x [\cos (\log_{t} x) - \sin (\log_{e} x)] + C

\frac{x}{2} [\cos (\log_{c} x) + \sin (\log_{e} x)] + C

x [\cos (\log_{i} x) + \sin (\log_{e} x)] + C

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Correct option is C

Let $I = \int \cos (\log_{e} x) d x$

$\begin{array}{l} \Rightarrow I = \cos (\log_{e} x) x + \int \sin (\log_{e} x) d x \\ \Rightarrow I = x \cos (\log_{e} x) + \sin (\log_{e} x) x - \int \cos (\log_{e} x) d x \end{array}$

$\Rightarrow I = \frac{x}{2} [\cos (\log_{e} x) + \sin (\log_{e} x)] + C$

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integral ∫cos⁡loge⁡xdx equals (where C is the constant of Integration)

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integral $\int \cos (\log_{e} x) d x$ equals (where $C$ is the constant of Integration)