# How do you integrate #int x^2 ln ^2x dx # using integration by parts?

We can integrate by parts using the logarithm as integral part, so that in the resulting integral we have a rational function:

Solve the resulting integral by parts again:

Substituting in the first expression:

and simplifying:

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To integrate ( \int x^2 \ln^2(x) , dx ) using integration by parts, we choose ( u = \ln^2(x) ) and ( dv = x^2 , dx ). Then, we find ( du ) and ( v ) as follows:

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

[ v = \int x^2 , dx = \frac{x^3}{3} ]

Now, we apply the integration by parts formula:

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

[ \int x^2 \ln^2(x) , dx = \frac{x^3}{3} \ln^2(x) - \int \frac{2x^3 \ln(x)}{3x} , dx ]

[ = \frac{x^3}{3} \ln^2(x) - \frac{2}{3} \int x^2 \ln(x) , dx ]

Now, we can solve the remaining integral ( \int x^2 \ln(x) , dx ) using integration by parts again or by using other methods like substitution.

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To integrate (x^2 \ln^2(x) , dx) using integration by parts, follow these steps:

- Identify the parts of the integrand to assign to (u) and (dv).
- Compute (du) and (v) by differentiating and integrating (u) and (dv), respectively.
- Apply the integration by parts formula: (\int u , dv = uv - \int v , du).
- Substitute the computed values into the formula and integrate.

Given (u = \ln^2(x)) and (dv = x^2 , dx), we can compute (du) and (v) as follows:

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

[ v = \int x^2 , dx = \frac{x^3}{3} ]

Now, apply the integration by parts formula:

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

[ = \frac{x^3}{3} \ln^2(x) - \int \frac{x^3}{3} \cdot \frac{2}{x} \ln(x) , dx ]

[ = \frac{x^3}{3} \ln^2(x) - \frac{2}{3} \int x^2 \ln(x) , dx ]

At this point, you may need to integrate (\int x^2 \ln(x) , dx) again using integration by parts or another method. However, it's worth noting that this integral does not have a simple closed-form solution and may involve special functions.

So, the final expression for (\int x^2 \ln^2(x) , dx) using integration by parts is:

[ \frac{x^3}{3} \ln^2(x) - \frac{2}{3} \int x^2 \ln(x) , dx ]

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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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