How do you integrate #(11x2) /(x^2 + x6)# using partial fractions?
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To integrate ( \frac{11x  2}{x^2 + x  6} ) using partial fractions, follow these steps:

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

Decompose the rational function into partial fractions based on the factored form of the denominator. ( \frac{11x  2}{x^2 + x  6} = \frac{A}{x + 3} + \frac{B}{x  2} )

Clear the denominators by multiplying both sides of the equation by ( (x + 3)(x  2) ): ( 11x  2 = A(x  2) + B(x + 3) )

Expand and collect like terms: ( 11x  2 = Ax  2A + Bx + 3B )

Match coefficients of corresponding terms on both sides of the equation: ( A + B = 11 ) (coefficients of ( x )) ( 2A + 3B = 2 ) (constant terms)

Solve the system of equations to find the values of ( A ) and ( B ).
From the first equation: ( A = 11  B )
Substitute into the second equation: ( 2(11  B) + 3B = 2 )
Simplify and solve for ( B ): ( 22 + 2B + 3B = 2 ) ( 5B = 20 ) ( B = 4 )
Then, ( A = 11  4 = 7 ).

Now that you have found the values of ( A ) and ( B ), rewrite the original expression using the partial fraction decomposition: ( \frac{11x  2}{x^2 + x  6} = \frac{7}{x + 3} + \frac{4}{x  2} )

Integrate each term separately: ( \int \frac{7}{x + 3} , dx = 7 \lnx + 3 + C ) ( \int \frac{4}{x  2} , dx = 4 \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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