Solve \int _1^e\left(\frac{\ln \left(x\right)}{x}\rig

Evaluate

\int_{1}^{e} \frac{ln ( x )}{x} d x

Evaluate the integral

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

Use the substitution d x = x d t to transform the integral

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\int \frac{ln ( x )}{x} \times x d t

Simplify

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\int ln (x) d t

Use the substitution t = ln (x) to transform the integral

\int t d t

Use the property of integral \int x^{n} d x = \frac{x ^{n + 1}}{n + 1}

\frac{t ^{1 + 1}}{1 + 1}

Add the numbers

\frac{t ^{2}}{1 + 1}

Add the numbers

\frac{t ^{2}}{2}

Substitute back

\frac{( ln ( x ) ) ^{2}}{2}

Simplify the expression

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

Return the limits

(\frac{1}{2} (ln (x))^{2})_{1}^{e}

Solution

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\frac{1}{2}

Alternative Form

0.5