A question about convergence of improper parametric integral

Question

Could you give me some hint how to find all $\alpha\in R$ for with the integral $\int_0^1 \frac{a-x^{\alpha}}{1-x}$ converges. Is clear that this integral converges for all$\alpha\in N$, but I could not understand how to deal in case $\alpha \notin N$.

Thanks.

Answer 1

By change of variable $t=1-x$ the integral becomes:

$$\int_0^1\frac{a-(1-t)^\alpha}{t}dt$$ and clearly if $a\ne1$ then $$\frac{a-(1-t)^\alpha}{t}\sim_0\frac{a-1}{t}$$ so the integral is divergent. Now let $\boxed{a=1}$ then $$\frac{a-(1-t)^\alpha}{t}\sim_0 \frac{\alpha t}{t}=\alpha$$ so the function can be extended by continuity on $0$ and then the integral is convergent.