Integrate the natural logarithm of x (ln x) with respect to x
In order to integrate ln x you have to use integration by parts, even though it appears there is only one term to be integrated. We get around this by instead writing it as (ln x)(1), where we treat the 1 as another function of x. Now we can apply the integration by parts rule by setting u = ln x and dv/dx = 1.
Integration by parts states that the integral of u(dv/dx) = uv - the integral of v(du/dx). Integrating v(du/dx) is easy because we know that d/dx(ln x) = 1/x, and the integral of 1 is x, so the two cancel and we are left with integrating 1 again. Once integrated fully, the answer will be x[(ln x) - 1] (+c) where c is the constant of integration.
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