Aus Online Mathematik Brückenkurs 2

We can discern two factors in the integrand, \displaystyle x and \displaystyle \ln x. 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 \displaystyle x because then it will become equal to 1, but then we have the problem of determining a primitive function for \displaystyle \ln x (how is that done?). Instead, a more successful way is to integrate \displaystyle x and to differentiate \displaystyle \ln x,

\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.