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Antiderivatives of Logarithmic Functions

Som formulas for computing antiderivatives of logarithmic functions are deduced.

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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