Fundamental Theorem of Calculus with 1/lnx

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I'm struggling with this problem, because I'm not sure how to integrate $1/\ln(x)$

Suppose that you have the following information about a function $F(x)$:

$$F(0)=1, F(1)=2, F(2)=5$$ $$F'(x)=\frac1{\ln(x)}$$

Using the Fundamental Theorem of Calculus evaluate $$\int_0^2 \frac2{\ln(x)}$$

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2 Answers

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You do not need to know how to integrate $1/\ln x$ (which is good, because it doesn't have an elementary antiderivative). You do need to know the numerical values of an antiderivative of $1/\ln x$ at the endpoints of the interval.

Do you know them?

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If $F'$ were continous on $[0,2]$ we had $$ I = 2 \int\limits_0^2\!\! F'(x) \, dx = 2 (F(2) - F(0)) = 2(5-1) = 8 $$ Alas this problem seems buggy, because of the singularity at $x=1$.

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