Antiderivatives of Logarithmic Functions

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

In this case, the technique of integration by parts is required to justify the formula. Define $u = \ln |\theta|$ and $dv = \theta\cdot d\theta$ . Therefore, $du = \nicefrac{d\theta}{\theta}$ and $v = \nicefrac{\theta^2}{2}$ .

Substitute this into the formula to compute the antiderivative using integration by parts:

$\begin{eqnarray*} \int\! \theta\cdot\ln|\theta|\cdot d\theta &=& \frac{\theta^2}{2} \cdot \ln |\theta| - \int\! \frac{\theta^2}{2} \cdot \frac{d\theta}{\theta} \\ &=& \frac{1}{2}\,\theta^2\,\ln|\theta| - \frac{1}{2}\int\! \theta \cdot d\theta\\ &=& \frac{1}{2}\,\theta^2\,\ln|\theta| - \frac{1}{4}\,\theta^2 + C \end{eqnarray*}$

Therefore, the general rule to compute the antiderivative is:

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

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Som formulas for computing antiderivatives of logarithmic functions are deduced.

Antiderivative of the Function

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