How do you evaluate the integral int 1/x dx from 0 to 1 if it converges?

Jul 17, 2016

${\int}_{0}^{1} \frac{1}{x} \mathrm{dx}$ diverges to $\infty$

Explanation:

As $\frac{1}{x} \to \infty$ as $x \to 0$, we must use an improper integral.

${\int}_{0}^{1} \frac{1}{x} \mathrm{dx} = {\lim}_{M \to 0} {\int}_{M}^{1} \frac{1}{x} \mathrm{dx}$

$= {\lim}_{M \to 0} {\left[\ln \left(x\right)\right]}_{M}^{1}$

$= {\lim}_{M \to 0} \left(\ln \left(1\right) - \ln \left(M\right)\right)$

$= {\lim}_{M \to 0} - \ln \left(M\right)$

$= \infty$

Thus, the integral diverges to infinity.