Zero Is Not In Its Domain

Calculus Level 3

Which of the following is true about the following definite integral?

∫ 0 1 ln ⁡ x d x \int _0 ^1 \ln x \, dx

It is equal to zero It is not defined, since ln ⁡ 0 \ln 0 is not defined It is a positive real number It is a negative real number It diverges to − ∞ - \infty , since lim ⁡ x → 0 + ln ⁡ x = − ∞ \displaystyle \lim _{x \to 0^+} \ln x = - \infty

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Pranshu Gaba by Pranshu Gaba , 22, India

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1 solution

Chew-Seong Cheong
May 11, 2016

Relevant wiki: Integration of Logarithmic Functions

∫ 0 1 ln ⁡ x d x = x ln ⁡ x − x ∣ 0 1 = ln ⁡ 1 − 1 − lim ⁡ x → 0 ( x ln ⁡ x ) + 0 = − 1 + lim ⁡ x → 0 ln ⁡ x 1 x Applying l’H o ˆ pital’s rule = − 1 + lim ⁡ x → 0 1 x − 1 x 2 = − 1 + lim ⁡ x → 0 ( − x ) = − 1 − 0 = − 1 \begin{aligned} \int_0^1 \ln x \ dx & = x \ln x - x \big|_0^1 \\ & = \ln 1 - 1 - \color{#3D99F6}{\lim_{x \to 0} (x \ln x)} + 0 \\ & = -1 + \color{#3D99F6}{\lim_{x \to 0} \frac{\ln x}{\frac{1}{x}} \quad \quad \small \text{Applying l'Hôpital's rule}} \\ & = -1 + \color{#3D99F6}{\lim_{x \to 0} \frac{\frac{1}{x}}{ -\frac{1}{x^2}}} \\ & = -1 + \color{#3D99F6}{\lim_{x \to 0} (-x)} \\ & = -1 \color{#3D99F6}{- 0} \\ & = \boxed{-1} \end{aligned}

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Moderator note:

LIATE is a really useful menmonic to do integration by parts!

thats wounderfull we can take limit f(x) if y dont exits when x =0 but thise take time true 100%

Patience Patience - 5 years, 1 month ago

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