1.54 Example of an improper integral

As $x\rightarrow 0+$, the integrand $x^{-s}\rightarrow\infty$, so the integral is improper.

Solution. For any $s\neq 1$ we have $\int_{\delta}^{1}x^{-s}\,dx={\left[\frac{1}{1-s}x^{1-s}\right]_{\delta}^{1}}\,{=\frac{1}{1-s}(1-\delta^{1-s})}$. If $s\in(0,1)$ then $\delta^{1-s}\rightarrow 0$ as $\delta\rightarrow 0+$. Thus the integral converges to $\frac{1}{1-s}$.

On the other hand, if $s>1$ then ${\delta^{1-s}\rightarrow\infty}$ as $\delta\rightarrow 0+$.

Finally, if $s=1$ then $\int_{\delta}^{1}\frac{1}{x}\,dx\,{=\left[\log x\right]_{\delta}^{1}}\,{=-\log\delta\rightarrow\infty}$ as $\delta\rightarrow 0+$.