How do you find #int ( 3x1)/(x^2+2x8) 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 stepbystep breakdown:

Factor the denominator: [ x^2 + 2x  8 = (x + 4)(x  2) ]

Decompose the fraction: [ \frac{3x  1}{x^2 + 2x  8} = \frac{A}{x + 4} + \frac{B}{x  2} ]

Solve for coefficients ( A ) and ( B ): [ 3x  1 = A(x  2) + B(x + 4) ]

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 ]

Integrate each partial fraction: [ \int \frac{1}{x + 4} , dx = \lnx + 4 + C_1 ] [ \int \frac{2}{x  2} , dx = 2\lnx  2 + C_2 ]

Combine the integrals: [ \int \frac{3x  1}{x^2 + 2x  8} , dx = \lnx + 4 + 2\lnx  2 + C ]
Where ( C ) is the constant of integration.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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