The value of $\int \cos \left(\log _e(x)\right) \mathrm{d} x$ is equal to (where $C$ is a constant of…

The value of $\int \cos \left(\log _e(x)\right) \mathrm{d} x$ is equal to (where $C$ is a constant of integration.)
  1. $x[\cos (\log x)-\sin (\log x)]+C$
  2. $\frac{x}{2}[\sin (\log x)-\cos (\log x)]+C$
  3. $\frac{x}{2}[\sin (\log x)+\cos (\log x)]+C$
  4. $x[\cos (\log x)+\sin (\log x)]+C$

Solution

$\begin{aligned} & \int \cos \left(\log _{\mathrm{e}} x\right) \mathrm{d} x \operatorname{let}_{\log _{\mathrm{e}} x=t} \\ & \Rightarrow \mathrm{d} x=e^t \\ & \Rightarrow I=\int \cos t \cdot e^t \mathrm{~d} t=\cos t \cdot e^t+\sin t \cdot e^t-I \quad[\text { Integrating by parts] } \\ & \Rightarrow 2 I=e^t(\cos t+\sin t) \\ & \Rightarrow I=\frac{x}{2}\left\{\cos \left(\log _{\mathrm{e}} x\right)+\sin \left(\log _{\mathrm{e}} x\right)\right\}\end{aligned}$

Asked in: MHT CET 2022 (11 Aug Shift 1)

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