question_answer

Evaluate \[\int_{1}^{e}{\frac{\cos ({{\log }_{e}}x)}{x}\,dx.}\]

Answer:

Let \[l=\int_{1}^{e}{\frac{\cos \,({{\log }_{e}}x)}{x}}\,dx\] Put \[t=\log x\Rightarrow dt=\frac{1}{x}\,dx\] Also, when\[x=1,\]then \[t=\log 1=0\] And when \[x=e,\]then \[t=\log e=1\] \[\therefore \]\[l-\int_{0}^{1}{\cos t\,dt=[\sin t]_{0}^{1}}=\sin 1-\sin 0-\sin 1\]