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\( \sqrt{n}=\int_{0}^{1}(\log 1 / y) d y \) |
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Answer» \(\int_0^1 \log \left(\frac{1}{y}\right) dy\) \(= \int_0^1 (\log 1 - \log y) dy\) \(\left(∵ \log \left(\frac{A}{B}\right) = \log A - \log B \right)\) \(= - \int_0^1 \log y\; dy\) (∵ log 1 = 0) \(= - \left[\log y \int 1\;dy - \int \left(\frac{d}{dy}^{\log y}\int 1\;dy \right) dy \right]_0^1\) \(= - \left[y\log y - \int \left(\frac{1}{y} \times y\right) dy \right]_0^1\) \(= - \left[y\log y - \int dy \right]_0^1\) \(= - \left[y\log y - y \right]_0^1\) \(= - \left[1\log 1 - 1 - 0 + 0 \right]\) = - (- 1) = 1 = √1 (∵ log 1 = 0) By comparing, we get n = 1. |
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