\(\int \cos (\log x) d x=\)

Let \(\mathrm{I}=\int \cos (\log x) \mathrm{d} x\)
Put \(\log _{\mathrm{e}} x=\mathrm{t} \Rightarrow x=\mathrm{e}^{\mathrm{t}} \Rightarrow \mathrm{d} x=\mathrm{e}^{\mathrm{t}} \mathrm{dt}\)
\(\therefore I =\int \cos t \cdot e^t d t \)
\( =\cos t \cdot e^t-\int(-\sin t) \cdot e^t d t \)
\( =\cos t \cdot e^t+\left[\sin t \cdot e^t-\int \cos t \cdot e^t d t\right] \)
\( \therefore I =\cos t \cdot e^t+\sin t \cdot e^t-I+c_1 \)
\( \Rightarrow 2 I=\cos t \cdot e^t+\sin t \cdot e^t+c_1 \)
\( \Rightarrow I=\frac{x}{2}\left[\cos \left(\log _e x\right)+\sin \left(\log _e x\right)\right]\) \(+c, \text { where } c=\frac{c_1}{2}\)