Question:
The integral $\int \cos \left(\log _{\mathrm{e}} \mathrm{x}\right) \mathrm{dx}$ is equal to :
(where $\mathrm{C}$ is a constant of integration)
Correct Option: , 2
Solution:
$I=\int \cos (\ell n x) d x$
$I=\cos (\ln x) \cdot x+\int \sin (\ell n x) d x$
$\cos (\ell n x) x+\left[\sin (\ell n x) \cdot x-\int \cos (\ell n x) d x\right]$
$\mathrm{I}=\frac{\mathrm{x}}{2}[\sin (\ell \mathrm{nx})+\cos (\ell \mathrm{n} x)]+\mathrm{C}$