Question

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

• A. 0
• B. -1
• C. 2
• D. 1

Answer 1

Answer: (D)

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