# How do I find the integral #int(x*ln(x))dx# ?

Integration by parts will be used.

Recall the formula for the IBP, which is

Entering the IBP formula provides us with:

With the power rule, the solution can now be found with ease. Additionally, remember the integration constant:

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To find the integral of ( \int x \ln(x) , dx ), you can use integration by parts. Let ( u = \ln(x) ) and ( dv = x , dx ). Then, ( du = \frac{1}{x} , dx ) and ( v = \frac{1}{2}x^2 ).

Now apply the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

Substitute the values:

[ \int x \ln(x) , dx = \frac{1}{2}x^2\ln(x) - \int \frac{1}{2}x^2 \cdot \frac{1}{x} , dx ]

[ = \frac{1}{2}x^2\ln(x) - \frac{1}{2} \int x , dx ]

[ = \frac{1}{2}x^2\ln(x) - \frac{1}{4}x^2 + 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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