# How do you integrate #(lnx)^2#?

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To integrate ((\ln(x))^2), you can use integration by parts. Let (u = \ln(x)) and (dv = \ln(x) , dx). Then (du = \frac{1}{x} , dx) and (v = x\ln(x) - \int x \cdot \frac{1}{x} , dx). Simplify (v) to (v = x\ln(x) - \int dx) and integrate to get (v = x\ln(x) - x).

Now apply the integration by parts formula:

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

Substitute the values:

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

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

[ \int (\ln(x))^2 , dx = x\ln(x) - x - x + C ]

[ \int (\ln(x))^2 , dx = x\ln(x) - 2x + C ]

So, the integral of ((\ln(x))^2) is (x\ln(x) - 2x + 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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