# What is the integral of #ln(x)/x#?

Lets start by breaking down the function.

So we have the two functions;

Now we can make some substitutions to the original integral.

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The integral of ( \frac{\ln(x)}{x} ) is a special function known as the logarithmic integral function, denoted as ( \text{Li}(x) ). It cannot be expressed in terms of elementary functions like polynomials, exponentials, or trigonometric functions. Therefore, its integral does not have a simple closed-form expression. However, it can be represented using the logarithmic integral function as follows:

[ \int \frac{\ln(x)}{x} , dx = \text{Li}(x) + C ]

Where ( \text{Li}(x) ) is the logarithmic integral function, and ( C ) is the constant of integration.

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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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