# How do you evaluate the definite integral #int dx / ( x(sqrt(ln(x)))# from #[e, e^81]#?

16

just as an alternative approach,

if you recognise the pattern

then the integral is

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We want to find:

Rewrite the integral:

Which can be integrated using the rule:

So:

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To evaluate the definite integral ∫ dx / (x√(ln(x))) from [e, e^81], we first need to find the antiderivative of the integrand. Then, we can use the Fundamental Theorem of Calculus to evaluate the integral over the given interval.

Let u = ln(x), then du = (1/x) dx. Substituting u = ln(x) and du = (1/x) dx into the integral, we get:

∫ dx / (x√(ln(x))) = ∫ du / (√u)

This is a standard integral, which evaluates to 2√u + C, where C is the constant of integration.

Now, substituting back u = ln(x), we get:

2√(ln(x)) + C

Now, we evaluate this antiderivative at the upper and lower bounds of integration:

At x = e: 2√(ln(e)) = 2√(1) = 2

At x = e^81: 2√(ln(e^81)) = 2√(81) = 2 * 9 = 18

Now, subtract the value at the lower bound from the value at the upper bound to find the definite integral:

Definite Integral = 18 - 2 = 16.

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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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