Question

${\int }_1^3\frac{\cos \left(\log x\right)}{x}dx$ is equal to

• A. $1$
• B. $\cos \left(\log 3\right)$
• C. $\sin \left(\log 3\right)$
• D. $\frac{\pi }{4}$

Answer 1

Answer: (C)

$\sin \left(\log 3\right)$

${\int }_1^3\frac{cos\left(logx\right)}{x}dx$

Substituting $logx=t$

$dt=\frac{1}{x}dx$

$\therefore {\int }_{t_1}^{t_2}costdt$

Lim $:t=log1=0$ to $t=log3$

$\therefore {\int }_0^{log3}costdt\Rightarrow [sint{]}_0^{log3}=sinlog3-0$

$\therefore {\int }_1^3\frac{cos\left(logx\right)}{x}dx=sinlog3$