# 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,

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

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

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

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