How do you integrate #ln(x) / x# from 4 to infinity?

Answer 1

the integral does not converge.

for the basic integration, spot the pattern:

#d/dx (ln^alpha x) = alpha ln^(alpha - 1) x * 1/x#

So

#d/dx (color{red}{1/2} ln^2 x) = 1/2 * 2 ln x * 1/x = ln x * 1/x#

So

#int_4^oo \ ln(x) / x \ dx = int_4^oo \ d/dx (1/2 ln^2 x) \ dx#
#= lim_(t to oo) (1/2 ln^2 x)_4^t #

the problem being, the integral does not converge.

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Answer 2

To integrate ln(x) / x from 4 to infinity, you can use the method of integration by parts. Let's denote u = ln(x) and dv = 1/x dx. Then, du = 1/x dx and v = ln(x).

Applying the integration by parts formula, ∫u dv = uv - ∫v du, we get:

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

This simplifies to:

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

Rearranging terms, we have:

2∫ln(x) / x dx = ln(x) * ln(x).

Now, solve for ∫ln(x) / x dx:

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

Now, to evaluate this from 4 to infinity:

lim (b→∞) ∫4 to b ln(x) / x dx = lim (b→∞) [(1/2) * ln(x) * ln(x)] from 4 to b.

= lim (b→∞) [(1/2) * ln(b) * ln(b)] - [(1/2) * ln(4) * ln(4)].

As b approaches infinity, ln(b) grows unbounded. Thus, the integral converges. Therefore, the integral from 4 to infinity of ln(x) / x dx equals infinity.

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Answer from HIX Tutor

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.

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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