Question

$\underset{1}{\overset{e^{\frac{17}{2}}}{\int }}\frac{\pi \cos (\pi \log x)}{x}dx=$

• A. 0
• B. -1
• C. 2
• D. 1

Answer 1

Answer: (D)

Let $I=\underset{1}{\overset{e^{\frac{17}{2}}}{\int }}\frac{\pi \cos (\pi \log x)}{x}dx$
Putting $pi\log x=z\Rightarrow \frac{\pi }{x}dx=dz$
Since, $1\le x\le e^{17/2}\Rightarrow 0\le z\le \frac{17\pi }{2}$
$\Rightarrow I={\int }_0^{17\pi /2}\cos zdx={\left[\sin z\right]}_0^{17\pi /2}$
$=\sin \frac{17\pi }{2}-\sin 0$
$=\sin \left(8\pi +\frac{\pi }{2}\right)=\sin \frac{\pi }{2}=1$