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a
$\frac{x}{2}\left[\mathrm{sin}\left({\mathrm{log}}_{e}x\right)-\mathrm{cos}\left({\mathrm{log}}_{c}x\right)\right]+C$
b
$x\left[\mathrm{cos}\left({\mathrm{log}}_{t}x\right)-\mathrm{sin}\left({\mathrm{log}}_{e}x\right)\right]+C$
c
$\frac{x}{2}\left[\mathrm{cos}\left({\mathrm{log}}_{c}x\right)+\mathrm{sin}\left({\mathrm{log}}_{e}x\right)\right]+C$
d
$x\left[\mathrm{cos}\left({\mathrm{log}}_{i}x\right)+\mathrm{sin}\left({\mathrm{log}}_{e}x\right)\right]+C$

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detailed solution

Correct option is C

Let $I=\int \mathrm{cos}\left({\mathrm{log}}_{e}x\right)dx$

$\begin{array}{l}⇒I=\mathrm{cos}\left({\mathrm{log}}_{e}x\right)x+\int \mathrm{sin}\left({\mathrm{log}}_{e}x\right)dx\\ ⇒I=x\mathrm{cos}\left({\mathrm{log}}_{e}x\right)+\mathrm{sin}\left({\mathrm{log}}_{e}x\right)x-\int \mathrm{cos}\left({\mathrm{log}}_{e}x\right)dx\end{array}$

$⇒I=\frac{x}{2}\left[\mathrm{cos}\left({\mathrm{log}}_{e}x\right)+\mathrm{sin}\left({\mathrm{log}}_{e}x\right)\right]+C$

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