Unveiling the Mystery of Improper Integrals: Why 1/x from 0 to 1 Diverges

Many students get tripped up by the integral of 1/x from 0 to 1. In this video, we'll use multiple graphs to show you why this integral doesn't exist. We'll explore the concept of improper integrals, vertical asymptotes, and why this particular function behaves so differently from others you might encounter in calculus. If you're struggling with this topic, this video is for you!

[@carultch]

Another interesting thing with this integral, is how it fits into the power rule in general. It turns out, with some clever assignments of the arbitrary constant of integration, we can show that it fills the hole where the power rule fails.
Given:
integral x^n dx = 1/(n + 1) * x^(n + 1) + C
Let n = -1 + h, where h is a very tiny number. Let C = -1/h.
1/(-1 + h + 1) * x^(-1 + h + 1) - 1/h
1/h * x^h - 1/h
1/h*(x^h - 1)
Take the limit as h approaches zero, using L'H's rule:
dN/dh = d/dh x^h = d/dh e^(ln(x)*h) = ln(x)*e^(ln(x)*h) = ln(x)*x^h
dD/dh = 1
dN/dh / (dD/dh) = ln(x)*x^h, evaluate at h=0, and get ln(x)
So you can see that ln(x) fills the hole where the integration power rule is degenerate.

[@ojaksteacher6893]

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