# How do you integrate #int 1/(x^2+x+1)# using partial fractions?

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To integrate ( \int \frac{1}{x^2+x+1} ) using partial fractions:

- Factor the denominator ( x^2 + x + 1 ) if possible. If it's irreducible, proceed to step 2.
- Express ( \frac{1}{x^2+x+1} ) as ( \frac{A}{x + \alpha} + \frac{B}{x + \beta} ) where ( \alpha ) and ( \beta ) are the roots of the quadratic equation ( x^2 + x + 1 = 0 ).
- Solve for ( A ) and ( B ) by multiplying both sides by the denominator ( x^2+x+1 ) and simplifying.
- After finding ( A ) and ( B ), rewrite ( \frac{1}{x^2+x+1} ) as ( \frac{A}{x + \alpha} + \frac{B}{x + \beta} ).
- Integrate each fraction separately.
- Finally, put the integral back together to get the solution.

This method allows you to integrate the given function using partial fractions.

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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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