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How do you integrate #1/ln(x) #?

Answer 1

Ezekiel Alexander

You don't. It is a special function called the logarithmic integral function.

You can find out more about the function here

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Answer 2

Axel Alvarez

To integrate ( \frac{1}{\ln(x)} ), you can use integration by parts.

Let ( u = \ln(x) ) and ( dv = dx ). Then, ( du = \frac{1}{x} , dx ) and ( v = x ).

Applying integration by parts:

[ \int \frac{1}{\ln(x)} , dx = x \ln(x) - \int x \frac{1}{x} , dx ]

Simplify:

[ \int \frac{1}{\ln(x)} , dx = x \ln(x) - \int dx ]

[ \int \frac{1}{\ln(x)} , dx = x \ln(x) - x + C ]

where ( C ) is the constant of integration.

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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.