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Aus Online Mathematik Brückenkurs 2

We can discern two factors in the integrand, x and lnx. If we are thinking about using partial integration, then one factor should be integrated and the other differentiated. It can seem attractive to choose to differentiate x because then it will become equal to 1, but then we have the problem of determining a primitive function for lnx (how is that done?). Instead, a more successful way is to integrate x and to differentiate lnx,

\displaystyle \begin{align} & \int{x\ln x\,dx=\frac{x^{2}}{2}\ln x}-\int{\frac{x^{2}}{2}}\centerdot \frac{1}{x}\,dx \\ & =\frac{x^{2}}{2}\ln x-\frac{1}{2}\int{x\,dx} \\ & =\frac{x^{2}}{2}\ln x-\frac{1}{2}\centerdot \frac{x^{2}}{2}+C \\ & =\frac{x^{2}}{2}\left( \ln x-\frac{1}{2} \right)+C \\ \end{align}

Thus, how one should the factors in a partial integration is very dependent on the situation and there are no simple rules.