log related integral

Calculus Level 2

evaluate ∫ 0 1 l n ( x ) d x \int_{0}^{1} ln(x) dx


The answer is -1.

Hemang Sarkar by hemang sarkar , 25, India

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1 solution

You can easily find the antiderivate of log ⁡ x \log x by using the integration by parts. ∫ log ⁡ x = x log ⁡ x − x + C \int \log x = x \log x - x + C

Taking the limits of integration: ∫ 0 1 log ⁡ x = x log ⁡ x − x ∣ 0 , 1 = − 1 − lim ⁡ x − > 0 x log ⁡ x \int\limits_0^1 \log x = x \log x - x |_{0, 1} = - 1 - \lim_{x -> 0} x \log x

Soving the limit by the L'hopital's rule: − 1 − lim ⁡ x − > 0 x log ⁡ x = − 1 − lim ⁡ x − > 0 log ⁡ x 1 x = − 1 − lim ⁡ x − > 0 − x 2 x = − 1 - 1 - \lim_{x -> 0} x \log x = - 1 - \lim_{x -> 0} \frac { \log x }{ \frac{1}{x} } = -1- \lim_{x -> 0} \frac {-x^2}{x} = -1

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Nice treatment of the limit

Josh Silverman Staff - 6 years, 10 months ago

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