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

Using integration by parts:

We let

This gives us:

Thus,

Use integration by parts one last time:

Thus,

Hence,

Don't forget the constant of integration!

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To integrate (ln x / 3)^3, you can use the substitution method. Let u = ln(x)/3. Then, du = (1/3x)dx. Rearrange to find dx = 3xdu. Substitute these into the integral and simplify. This yields the integral of u^3 * 3x du, which simplifies to 3∫(u^3 * x) du. Since u = ln(x)/3, x = e^(3u). Substitute x and du in terms of u into the integral, giving 3∫(u^3 * e^(3u)) du. Integrate this expression, and then resubstitute u = ln(x)/3 to obtain the final result.

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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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