Integral from 0 to 1 1/x

Question

I want to rewrite the integral $\int_{0}^{1} 1/x dx$ Using the substitution $\chi=1/x$ can this be written as: $\int_{1}^{\infty} 1/x dx$ Is this correct?

thanks

Answer 1

The integrals $\int_0^1\frac{1}{x}\,dx$ and $\int_1^\infty\frac{1}{x}\,dx$ are divergent.

However, for any $\epsilon>0$,

$$\int_{\epsilon}^1\frac{1}{x}\,dx=\int_{1}^{1/\epsilon}\frac{1}{x}\,dx$$

The equality follows from the suggested substitution $x \to 1/x$. We can in fact, carry out the integral(s) in $(1)$ and obtain

$$\int_{\epsilon}^1\frac{1}{x}\,dx=-\log \epsilon$$

and

$$\int_{1}^{1/\epsilon}\frac{1}{x}\,dx=\log (1/\epsilon)=-\log \epsilon$$

as expected!