# How do you find the integral from e^2 to e of #dx / (x * ln (x^8))#?

By applying the standard Log function rules, we determine that the Integrand is

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To find the integral ∫(dx / (x * ln(x^8))) from e^2 to e, you can use the substitution method. Let u = ln(x^8), then du = (8/x) dx. Rearranging this, we get dx = (x/8) du. Substituting these into the integral, we have ∫(dx / (x * ln(x^8))) = ∫(1/(8u)) du. Now, this integral becomes ∫(1/(8u)) du = (1/8) * ln|u| + C = (1/8) * ln|ln(x^8)| + C. To evaluate the definite integral from e^2 to e, plug in the upper and lower limits: (1/8) * [ln|ln(e^8)| - ln|ln(e^2)|] = (1/8) * [ln|8| - ln|2|] = (1/8) * (ln(8) - ln(2)) = (1/8) * ln(4) = ln(√2) / 4. Therefore, the integral from e^2 to e of dx / (x * ln (x^8)) equals ln(√2) / 4.

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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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