What is the integral of (ln((e^x)/x))/(x) ?

The integral of (ln((e^x)/x))/(x) is x-ln(x)-((ln(x)-1)^2)/2+C

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integral of (ln((e^x)/x))/(x)

\int \frac{ln ( \frac{e ^{x}}{x} )}{x} d x

x - ln (x) - \frac{( ln ( x ) - 1 ) ^{2}}{2} + C

Solution steps

\int \frac{ln ( \frac{e ^{x}}{x} )}{x} d x

Apply Integration By Parts

: x - ln (x) - \int \frac{ln ( x ) - 1}{x} d x

= x - ln (x) - \int \frac{ln ( x ) - 1}{x} d x

\int \frac{ln ( x ) - 1}{x} d x = \frac{( ln ( x ) - 1 ) ^{2}}{2}

= x - ln (x) - \frac{( ln ( x ) - 1 ) ^{2}}{2}

Add a constant to the solution

= x - ln (x) - \frac{( ln ( x ) - 1 ) ^{2}}{2} + C

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What is the integral of (ln((e^x)/x))/(x) ?
The integral of (ln((e^x)/x))/(x) is x-ln(x)-((ln(x)-1)^2)/2+C