How do you find #int ( 3x-1)/(x^2+2x-8) dx# using partial fractions?
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To find ( \int \frac{3x - 1}{x^2 + 2x - 8} , dx ) using partial fractions, follow these steps:
- Factor the denominator ( x^2 + 2x - 8 ) to identify its roots.
- Decompose the fraction into partial fractions with undetermined coefficients.
- Solve for the coefficients by equating the original expression with the decomposition.
- Integrate each partial fraction separately.
- Combine the integrals to find the final result.
Here's a step-by-step breakdown:
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Factor the denominator: [ x^2 + 2x - 8 = (x + 4)(x - 2) ]
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Decompose the fraction: [ \frac{3x - 1}{x^2 + 2x - 8} = \frac{A}{x + 4} + \frac{B}{x - 2} ]
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Solve for coefficients ( A ) and ( B ): [ 3x - 1 = A(x - 2) + B(x + 4) ]
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Equate coefficients: [ 3x - 1 = Ax - 2A + Bx + 4B ] [ (3 - A + B)x + (-2A + 4B) = 3x - 1 ]
Matching coefficients: [ 3 - A + B = 3 ] (for ( x )) [ -2A + 4B = -1 ] (constant term)
Solving these equations gives: [ A = 1 ] [ B = 2 ]
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Integrate each partial fraction: [ \int \frac{1}{x + 4} , dx = \ln|x + 4| + C_1 ] [ \int \frac{2}{x - 2} , dx = 2\ln|x - 2| + C_2 ]
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Combine the integrals: [ \int \frac{3x - 1}{x^2 + 2x - 8} , dx = \ln|x + 4| + 2\ln|x - 2| + 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.
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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