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

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Answer 1 · 2015-07-08T22:02:00

#int color(green)((1+lnx))xlnxdx#

If we differentiate #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)#