Antiderivatives of Logarithmic Functions

Antiderivative of the Function $y = v^n\,\ln v$

Define $u = \ln|\theta|$ and $dv = \theta^{n}\cdot d\theta$ , so that $du = \nicefrac{d\theta}{\theta}$ and $v = \nicefrac{\theta^{n+1}}{(n + 1)}$ .

Now apply the formula of Integration by parts,

$\begin{eqnarray*} \int\!\theta^{n}\cdot \ln|\theta|\cdot d\theta &=& \frac{\theta^{n+1}}{(n + 1)}\cdot\ln|\theta| - \int\! \frac{\theta^{n+1}}{(n + 1)}\cdot \frac{d\theta}{\theta}\\ &=& \frac{\theta^{n+1}}{(n + 1)}\cdot\ln|\theta| - \frac{1}{(n + 1)}\,\int\!\theta^{n}\cdot d\theta\\ &=& \frac{\theta^{n+1}}{(n + 1)}\cdot\ln|\theta| - \frac{1}{(n + 1)^2}\,\theta^{n+1} + C\\ &=& \frac{\theta^{n+1}}{(n + 1)^2}\cdot\left[(n + 1) \, \ln|\theta| - 1\right] + C \end{eqnarray*}$

And the rule to compute the antiderivative is generalized by means of the chain rule as:

$\begin{equation*} \int\!v^{n}\cdot \ln|v|\cdot dv = \frac{v^{n+1}}{(n + 1)^2}\cdot\left[(n + 1) \, \ln|v| - 1\right] + C \end{equation*}$

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Som formulas for computing antiderivatives of logarithmic functions are deduced.

Antiderivative of the Function

Antiderivative of the Function $y = v^n\,\ln v$