The integral ∫0115[1x]dx. Where [.] denotes the greatest integer function is equal to

∫01dx5[1x]=−∫10dx5[1x] =−[∫11/2dx5+∫1/21/3dx52+.....] =−log45−5[−log45−15]=1+4log45

The integral ∫0115[1x]dx. Where [.] denotes the greatest integer function is equal to

∫01dx5[1x]=−∫10dx5[1x] =−[∫11/2dx5+∫1/21/3dx52+.....] =−log45−5[−log45−15]=1+4log45

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The integral $\int_{0}^{1} \frac{1}{5^{[\frac{1}{x}]}}$ dx. Where [.] denotes the greatest integer function is equal to

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$1 + 4 \log (\frac{4}{5})$

$1 - 4 \log (\frac{4}{5})$

$\log_{e} (\frac{4}{5})$

$1 - 5 \log (\frac{4}{5})$

answer is A.

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Detailed Solution

$\int_{0}^{1} \frac{d x}{5^{[\frac{1}{x}]}} = - \int_{1}^{0} \frac{d x}{5^{[\frac{1}{x}]}} = - [\int_{1}^{1 / 2} \frac{d x}{5} + \int_{1 / 2}^{1 / 3} \frac{d x}{5^{2}} + .....] \begin{array}{l} = - \log \frac{4}{5} - 5 [- \log \frac{4}{5} - \frac{1}{5}] \\ = 1 + 4 \log \frac{4}{5} \end{array}$