How do you integrate $(1 + ln x) x ln x$ $(1+lnx)xlnx$ ?

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Answer 1 · 2015-07-08T22:02:00

$\int (1 + ln x) x ln x d x$ $int color(green)((1+lnx))xlnxdx$

If we differentiate $x ln x$ $xlnx$ , by the Product Rule, we get:

$= f(x)g'(x) + g(x)f'(x)$

$= x*1/x + lnx*1 = color(green)(1 + lnx)$

Thus, let:
$u = xlnx$
$du = (1+lnx)dx$

$= int udu$

$= u^2/2$

$= color(blue)((x^2ln^2x)/2 + C)$

How do you integrate (1+lnx)xlnx(1+lnx)xlnx(1+lnx)xlnx?