Which Integral Form Matches the Simplification of x(lnx)^2 dx?

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SUMMARY

The integral of x(lnx)^2 dx simplifies to (1/4)x^2(2(ln(x)-1)ln(x)+1). The correct form among the provided options is 1/2 x^2 ((lnx)^2 - lnx + 1/2) + C. This conclusion is reached by manipulating the original expression to match one of the specified integral forms. The transformation confirms that the integral can be expressed in a more concise format while maintaining equivalence.

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electronicxco
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I have found for the integral of x(lnx)^2 dx to be :
((1)/(4)x^2)(2(ln(x)-1)ln(x)+1)

but i need it to be in one of these forms:
these are the choices:
x^2((lnx)^2 + lnx- (1/2)) +C
1/2 x^2 ((lnx)^2 +lnx + 1/2) +C
x^2((lnx)^2 + lnx + (1/2)) + C
x^2((lnx)^2 - lnx + (1/2)) + C
1/2 x^2 ((lnx)^2 -lnx + 1/2) +C
1/2 x^2 ((lnx)^2 -lnx - 1/2) +C

Thanks!
 
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((1)/(4)x^2)(2(ln(x)-1)ln(x)+1) = (1/2)x^2[(ln(x) - 1)ln(x) + 1/2] = 1/2 x^2 ((lnx)^2 -lnx + 1/2), so

1/2 x^2 ((lnx)^2 -lnx + 1/2) +C

is the correct choice.
 

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