Question:
The integral $\int \cos \left(\log _{e} x\right) d x$ is equal to :
(where $\mathrm{C}$ is a constant of integration)
Correct Option: , 3
Solution:
Let the integral, $I=\int \cos (\ln x) d x$
$\Rightarrow \quad I=\cos (\ln x) \cdot x-\int \frac{-\sin (\ln x)}{x} x d x$
$=x \cos (\ln x)+\int \sin (\ln x) d x$
$=x \cos (\ln x)+\sin (\ln x) x-\int \frac{\cos (\ln x)}{x} x d x$
$=x \cos (\ln x)+\sin (\ln x) \cdot x-I$
$\Rightarrow \quad 2 I=x(\cos (\ln x)+\sin (\ln x))+C$
$\Rightarrow \quad I=\frac{x}{2}[\cos (\ln x)+\sin (\ln x)]+C$