How do you use partial fraction decomposition to decompose the fraction to integrate #11/(3x^27x6)#?
Factorize the denominator by solving the equation:
Then:
Apply partial fractions decomposition:
So:
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To decompose the fraction ( \frac{11}{3x^2  7x  6} ) using partial fraction decomposition:

Factor the denominator (3x^2  7x  6) into linear factors. (3x^2  7x  6 = (3x + 2)(x  3))

Write the fraction with undetermined coefficients. ( \frac{11}{3x^2  7x  6} = \frac{A}{3x + 2} + \frac{B}{x  3} )

Clear the fractions by multiplying both sides by the denominator. (11 = A(x  3) + B(3x + 2))

Solve for the unknown coefficients (A) and (B) by equating coefficients of like terms. (11 = Ax  3A + 3Bx + 2B)

Group like terms and equate coefficients. (11 = (A + 3B)x + (3A + 2B))

Equate coefficients: (A + 3B = 0) (for the term with (x)) (3A + 2B = 11) (for the constant term)

Solve the system of equations to find the values of (A) and (B). Solving the system gives (A = \frac{1}{7}) and (B = \frac{4}{7}).

Substitute the values of (A) and (B) back into the original fraction.
[ \frac{11}{3x^2  7x  6} = \frac{1}{7} \cdot \frac{1}{3x + 2} + \frac{4}{7} \cdot \frac{1}{x  3} ]
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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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