# How do you integrate #int x(lnx)^2 # using integration by parts?

The answer is

The integration by parts is

Here,

Therefore, the integral is

Therefore,

Putting all together

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To integrate ( \int x(\ln x)^2 ) using integration by parts, you would use the formula:

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

Let ( u = (\ln x)^2 ) and ( dv = x , dx ).

Then, differentiate ( u ) to find ( du ) and integrate ( dv ) to find ( v ).

[ du = 2\ln x \cdot \frac{1}{x} , dx = \frac{2}{x} \ln x , dx ] [ v = \frac{x^2}{2} ]

Now, apply the integration by parts formula:

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

Simplify and integrate the remaining term:

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

This remaining integral can be further simplified and evaluated using integration by parts or other methods if necessary.

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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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