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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 integration by parts, 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,

xlnxdx=2x2lnx−2x2x1dx=2x2lnx−21xdx=2x2lnx−212x2+C=2x2lnx−21+C.

Thus, how one should choose the factors in an integration by parts is very dependent on the situation and there are no simple rules.