What is the antiderivative of # (xlnx - x) #?

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Answer 1 · 2016-11-23T04:45:01

#1/2x^2lnx-3/4x^2+C#

This is the same as asking

#int(xlnx-x)dx#

Splitting up the integral:

#=intxlnxdx-intxdx#

The second can be integrated using the power rule for integration:

#=intxlnxdx-x^2/2#

For the remaining integral, use integration by parts. This takes the form #intudv=uv-intvdu#. For #intxlnxdx#, let:

#{(u=lnx,=>,du=1/xdx),(dv=xdx,=>,v=x^2/2):}#

Thus:

#=uv-intvdu-x^2/2#

#=1/2x^2lnx-1/2intx^2/xdx-x^2/2#

#=1/2x^2lnx-1/2x^2/2-x^2/2#

#=1/2x^2lnx-3/4x^2+C#