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:

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

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

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

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

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