# What is the integral of #xln(x)#?

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The integral of ( x \ln(x) ) with respect to ( x ) can be found using integration by parts. Applying integration by parts, we let ( u = \ln(x) ) and ( dv = x , dx ). Then, ( du = \frac{1}{x} , dx ) and ( v = \frac{x^2}{2} ).

[ \begin{align*} \int x \ln(x) , dx &= x \cdot \frac{x^2}{2} - \int \frac{x^2}{2} \cdot \frac{1}{x} , dx \ &= \frac{x^3}{2} - \frac{1}{2} \int x , dx \ &= \frac{x^3}{2} - \frac{1}{2} \cdot \frac{x^2}{2} + C \ &= \frac{x^3}{2} - \frac{x^2}{4} + C \end{align*} ]

Therefore, the integral of ( x \ln(x) ) with respect to ( x ) is ( \frac{x^3}{2} - \frac{x^2}{4} + C ), where ( C ) is the constant of integration.

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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.

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.

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.

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