# How do you integrate #(ln(x)*(1/x))dx#?

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To integrate ln(x)*(1/x) dx, you can use integration by parts. Let u = ln(x) and dv = (1/x) dx. Then, differentiate u to get du = (1/x) dx and integrate dv to get v = ln(x).

Applying the integration by parts formula:

∫u dv = uv - ∫v du

This yields:

∫ln(x)*(1/x) dx = ln(x) * ln(x) - ∫ln(x) * (1/x) dx

Now, the integral on the right side is the same as the original integral, so we can rewrite it as:

∫ln(x)*(1/x) dx = ln(x) * ln(x) - ∫ln(x)*(1/x) dx

Adding ∫ln(x)*(1/x) dx to both sides:

2∫ln(x)*(1/x) dx = ln(x) * ln(x)

Now, solve for ∫ln(x)*(1/x) dx:

∫ln(x)*(1/x) dx = (1/2) * ln(x) * ln(x) + 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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