$$\int_0^1 \frac{x-\ln(1+x)}{x^p}\,dx$$
Find the values of $p > 0$ so that the integral converges.
I don't have any idea.
How to think about it?
The answer is $0<p<3$
2026-03-27 12:34:18.1774614858
Find values of $p>0$ so that the integral converges.
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Use the comparison test and $$ \ln(1+x)=x-\frac{x^2}{2}+O(x^3). $$ As $x\to0$, $$ \frac{x-\ln(1+x)}{x^p}=x^{2-p}+O(x^{3-p}). $$ The integral behaves like$$ \int_0^1x^{2-p}\,dx, $$ which is convergent if $2-p>-1$.