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

integral $\int \mathrm{cos}\left({\mathrm{log}}_{e}x\right)dx$ equals (where is the constant of Integration)

1. A

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

2. B

$x\left[\mathrm{cos}\left({\mathrm{log}}_{t}x\right)-\mathrm{sin}\left({\mathrm{log}}_{e}x\right)\right]+C$

3. C

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

4. 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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Solution:

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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