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

Answer 1

Emma Aldridge

#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)#

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Answer 2

Isaiah Allen

To integrate ( (1 + \ln(x))x\ln(x) ), you can use integration by parts. The result of the integration is ( \frac{1}{2}x^2(\ln(x))^2 + \frac{1}{4}x^2 + \frac{1}{4}x^2\ln(x) + C ), where ( C ) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.