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  4. ∫ limits0 e37(π sin(π log x)/x)dx is equal to

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Q. $ \int\limits_{0} ^{_e37}\frac{\pi\,\sin(\pi\,\log\,x)}{x}dx$ is equal to

Integrals

Solution:

Put $\pi\, log\,x=z$
$\therefore \frac{\pi}{x} dx=dz$
When $x= 1, z = 0$, When $x =e^{37}, \pi\, log\, e^{37}=z$
$\therefore z=37\,\pi$
$\therefore $ given integral $=\int^{37\pi}_{0} sin\,z\, dz =\left|-cos\,z\right|_{0}^{37 \pi}$
$=-cos\left(37\,\pi\right)+cos\,0=1-cos\,\pi =1+1=2$