Dr. Fiendish 2

Calculus Level 1

Dr. Fiendish has a hard time teaching his pig, Freddy, to learn calculus. Today he gave it an exercise: to find
∫ − 1 1 log ⁡ 10 x x − 2 d x . \int_{-1}^1\frac{\log_{10} x}{x-2}dx. Can you help it with the problem?

P.S. No complex values allowed.

No, because this problem is flawed. Yes, this is just simple calculus.

Jeff Giff by Jeff Giff , 13, China

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2 solutions

Jeff Giff
Jun 7, 2020

At first glance, the denominator of this function is not 0 on the interval [ 1 , − 1 ] [1,-1] , but the numerator includes a logarithm , i.e. this equation is NOT defined on the interval [ 0 , − 1 ] [0,-1] , making this problem unsolvable.

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The value of the integral is not real, but complex . If complex values are allowed, then there is no flaw in the problem, but otherwise it is flawed. [The value of the integral involves ln ⁡ ( − 1 2 ) \ln (-\frac{1}{2}) ].

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Oh, I’ll fix that. Thanks a lot!

Jeff Giff - 1 year ago

The complex value of natural logarithm depends on the analytical continuation of the known domain to the complex axis.Example,ln(-1)=π(√-1)

Aruna Yumlembam - 1 year ago

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Yes, so I wrote: no complex values allowed. BTW, thx for reminding me! :)

Jeff Giff - 1 year ago

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