Indefinite Integration 1 Question 56

57. The value of $\int _1^{37 \pi} \frac{\pi \sin (\pi \log x)}{x} d x$ is

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Solution:

  1. Let $I=\int _1^{37 \pi} \frac{\pi \sin (\pi \log x)}{x} d x$

Put $\quad \pi \log x=t$

$$ \Rightarrow \frac{\pi}{x} d x=d t $$

$\therefore \quad I=\int _0^{37 \pi} \sin (t) d t=-[\cos t] _0^{37 \pi}=-[\cos 37 \pi-\cos 0]$

$$ =-[(-1)-1]=2 $$



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