# How can i integrate #1/(x^8-x)#?

We start by transforming the integrand into something more integrable.

So

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Hence:

So:

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To integrate ( \frac{1}{x^8 - x} ), you can use partial fraction decomposition followed by integration. The first step is to factor the denominator, which is ( x(x^7 - 1) ). Then, you decompose ( \frac{1}{x^8 - x} ) into partial fractions. After decomposing, integrate each term separately.

The decomposition will have the form:

[ \frac{1}{x(x^7 - 1)} = \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x - \omega} + \frac{D}{x - \omega^2} + \ldots + \frac{H}{x - \omega^7} ]

Where ( \omega ) is a complex number, a primitive 8th root of unity.

Once you find the constants ( A, B, C, \ldots, H ), integrate each term separately. This integration might involve logarithms and other elementary functions depending on the specific values of the constants.

However, note that integrating rational functions with high-degree polynomials in the denominator can be complex, and the resulting integrals may not have elementary solutions in terms of standard functions.

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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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