# How do you integrate #ln(x)/x^3#?

So essentially we are looking for one function that simplifies when it is differentiated, and one that simplifies when integrated (or at least is integrable).

So IBP gives;

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To integrate ln(x)/x^3, you can use integration by parts. Let u = ln(x) and dv = 1/x^3 dx. Then differentiate u to get du = (1/x) dx, and integrate dv to get v = -1/(2x^2).

Now apply the integration by parts formula:

∫u dv = uv - ∫v du

Substituting the values we found:

∫ln(x)/x^3 dx = -ln(x)/(2x^2) - ∫(-1/(2x^2))(1/x) dx

Simplify the integral:

∫ln(x)/x^3 dx = -ln(x)/(2x^2) + 1/(2x^3) ∫ dx

Integrate 1/(2x^3) to get (-1/(4x^2)), and combine terms:

∫ln(x)/x^3 dx = -ln(x)/(2x^2) - 1/(4x^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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