# Finding $\int_0^1 \frac{1}{x}\; dx$ using definition

I believe I understand how to find definite integrals of polynomials using the limit definition. So, for example, to find $\int_0^1 x^2$ one ends up needing $\sum_{i=1}^n i^2$. My question is how one would do an integral like $$\int_0^1 \frac{1}{x}\;dx$$ using the definition. From my work I end up having to find $$\sum_{i=1}^n \frac{1}{i}$$ and I don't know this sum.

• en.wikipedia.org/wiki/Harmonic_number Oct 31, 2017 at 19:04
• It diverges....
– user223391
Oct 31, 2017 at 19:05
• You can't integrate from $0$.
– user65203
Oct 31, 2017 at 19:11

I think you're talking about using the Riemann sum to calculate integrals. For example, $$\int_0^1x^2dx=\lim_{n\rightarrow\infty}\sum_{i=1}^n\left(\frac{i}{n}\right)^2\frac{1}{n}=\lim_{n\rightarrow\infty}\frac{1}{n^3}\sum_{i=1}^ni^2=\lim_{n\rightarrow\infty}\frac{n(n+1)(2n+1)}{6n^3}=\frac{1}{3}.$$
But the integral $$\int_0^1\frac{dx}{x}=\lim_{n\rightarrow\infty}\sum_{i=1}^n\frac{n}{i}\frac{1}{n}=\sum_{i=1}^\infty\frac{1}{i}=\infty$$