Evaluate using substitution of \int_{e^{2}}^{e^{3}} \frac{d x}{x \left ( \ln{x} - 1 \right )}

Solve int_e^{2}^e^{3} fracd xx ( ln{x - 1 )}

Evaluate

\int_{e^{2}}^{e^{3}} \frac{1}{x ( ln ( x ) - 1 )} d x

Multiply the terms

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\int_{e^{2}}^{e^{3}} \frac{1}{x ln ( x ) - x} d x

Evaluate the integral

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

Rewrite the expression

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

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

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

Simplify

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

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

\int \frac{1}{t - 1} d t

Use the property of integral \int \frac{1}{a x + b} d x = \frac{1}{a} ln ∣ a x + b ∣

ln (∣ t - 1 ∣)

Substitute back

ln (∣ ln (x) - 1 ∣)

Return the limits

(ln (∣ ln (x) - 1 ∣))_{e^{2}}^{e^{3}}

Solution

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ln (2)

Alternative Form

\approx 0.693147