How do you integrate the natural logarithm?
We already know that the derivative of ln(x) is 1/x, so how can this be used for integration?Recall integration by parts uses knowledge of the derivative of one of the parts in the formula.int(u v' dx) = uv - int(u' v dx)We now set the integrand to ln(x) * 1 and set u = ln(x) and v' = 1, so now we only need to differentiate ln(x), which we know how to do.Now differentiate u and integrate v' to get u' and v: u' = 1/x and v = x.Finally we can plug the results into the integration by parts formula:int(ln(x) dx) = ln(x)*x - int(1/x * x dx)int(ln(x) dx) = xln(x) - int(1 dx)int(ln(x) dx) = xln(x) - x + C
JL
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