How do I integrate ln(x)?

There is a subtle, but very neat trick to this when applying the rules of integration by parts.

If we take ∫ln(x)dx = ∫1*ln(x)dx, and then let our term to be differentiated, u = ln(x), and our term to be integrated, dv/dx = 1, then it follows that:


du/dx = x⁻¹, v = x


and from the integration by parts formula:

∫u * (dv/dx) dx = uv - ∫v * (du/dx) dx


∴ ∫ln(x)dx = xln(x) - ∫(x⁻¹ * x)dx (+ constant)

∫ln(x)dx = xln(x) - ∫dx (+ constant)

Hence, our results turns out to be:

∫ln(x)dx = xln(x) - x + c


NB. While our trick here gives us a very straightforward solution to an integration which could have been very laborious via other methods, integration by parts tends to be a last resort, as more, seemingly contrived steps are required. One should generally try integration by substitution, for non-standard integrations, first when unsure of which method to use, as the steps to a result are often far simpler and quicker.

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