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I have been asked to find the domain of the function $$f:x\mapsto \int_0^1\frac{t^x-1}{\ln(t)}dt$$ where $ x $ is real.

Obviously, $ f(0) $ exists, but for $ x\ne 0$ i couldn't find the criteria to use.

hamam_Abdallah
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    Technically, "what is the domain" is not the right question. What you really want to know is, for what (real) values of $x$ does the integral converge? – Robert Israel Mar 23 '22 at 18:01

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Evaluate $$ I=\int_0^1 \frac{t^x -1}{\log(t)}dt $$ using Feynman's trick. $$ I_{\alpha}=\int_0^1 \frac{t^{\alpha} -1}{\log(t)}dt $$ $$ \frac{dI_{\alpha}}{d\alpha}=\int_0^1 \frac{d}{d\alpha} \frac{e^{\alpha \log(t)} -1}{\log(t)}dt=\int_0^1 t^{\alpha}dt =\frac{1}{\alpha +1} $$ Thus, $$ I_{\alpha}= \log(\alpha +1) + C $$ where $C$ is the constant of integration. Setting $\alpha =0$ we find $C=0$ and $$ I=\log(x +1) $$ For real $I$ we must have $x \gt -1$

F. Wright
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