# How do you use partial fractions to find the integral #int 3/(x^2+x-2) dx#?

The integral equals

Thus , our partial fraction decomposition will be of the form

Write a system of equations and solve:

Therefore, the integral becomes

Hopefully this helps!

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To integrate the function ( \frac{3}{x^2 + x - 2} ) using partial fractions, first factor the denominator as ( (x + 2)(x - 1) ). Then, express ( \frac{3}{x^2 + x - 2} ) as the sum of two fractions with undetermined coefficients, ( \frac{A}{x + 2} ) and ( \frac{B}{x - 1} ).

Solve for ( A ) and ( B ) by equating the original expression to the partial fraction decomposition. This leads to ( A = 1 ) and ( B = 2 ).

Therefore, the integral becomes ( \int \frac{3}{x^2 + x - 2} dx = \int \frac{1}{x + 2} dx + \int \frac{2}{x - 1} dx ).

Now, integrate each term separately to get ( \ln|x + 2| + 2\ln|x - 1| + C ), where ( 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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