Logarithmic Functions | Fundamental Integration Formulas

The limitation of the Power Formula $ \displaystyle \int u^n \, du = \dfrac{u^{n + 1}}{n + 1} + C $, is when $ n = -1 $; this makes the right side of the equation indeterminate. This is where the logarithmic function comes in, note that $ \displaystyle \int u^{-1} \, du = \displaystyle \int \frac{du}{u} $, and we can recall that $ d(\ln u) = \dfrac{du}{u} $. Thus,
 

$ \displaystyle \int \dfrac{du}{u} = \ln u + C $

 

The formula above involves a numerator which is the derivative of the denominator. The denominator $ u $ represents any function involving any independent variable. The formula is meaningless when $ u $ is negative, since the logarithms of negative numbers have not been defined. If we write $ u = -v $ so that $ du = -dv $, then we have
 

$ \displaystyle \int \dfrac{du}{u} = \int \dfrac{-dv}{-v} = \int \dfrac{dv}{v} = \ln v + C = \ln (-u) + C $

 
When negative numbers are involved, the formula should be considered in the form
 

$ \displaystyle \int \dfrac{du}{u} = \ln | u | + C $

 
The integral of any quotient whose numerator is the differential of the denominator is the logarithm of the denominator.
 

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