# How do you find the integral #int_e^(e^4)dx/(x*sqrt(ln(x)))dx# ?

To find the integral of your function, we do the following:

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To find the integral ( \int_{e}^{e^4} \frac{dx}{x \sqrt{\ln(x)}} ), you can use the substitution method. Let ( u = \ln(x) ), then ( du = \frac{1}{x} dx ). Substituting this into the integral, we get:

[ \int \frac{dx}{x \sqrt{\ln(x)}} = \int \frac{1}{\sqrt{u}} du ]

Now, the integral becomes:

[ \int \frac{1}{\sqrt{u}} du ]

This is a standard integral which can be easily evaluated. Its antiderivative is ( 2\sqrt{u} ). Substituting back ( u = \ln(x) ), we get:

[ 2\sqrt{\ln(x)} + C ]

So, the integral ( \int_{e}^{e^4} \frac{dx}{x \sqrt{\ln(x)}} ) evaluates to ( 2\sqrt{\ln(e^4)} - 2\sqrt{\ln(e)} ) which simplifies to ( 2\sqrt{4} - 2\sqrt{1} = 4 - 2 = 2 ).

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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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