How do you integrate #x^2/( (x1)(x+2))# using partial fractions?
As the numerator has the same degree of the denominator we start by separating the rational function in a way that allows a simplification:
The first addendum can be separated again to simplify:
Now we write the second term (in red) as a sum of partial fractions with parametric numerators:
compute the sum in the second member:
As the denominators are equal, the equation is satisfied when the numerators are equal:
which implies:
Substituting in the expressions above:
We are now ready to integrate:
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To integrate ( \frac{x^2}{(x1)(x+2)} ) using partial fractions, follow these steps:

Factor the denominator: ( (x1)(x+2) ) factors into ( x^2 + x  2 ).

Decompose the fraction into partial fractions: ( \frac{x^2}{(x1)(x+2)} = \frac{A}{x1} + \frac{B}{x+2} )

Clear the fractions by multiplying both sides by the common denominator ( (x1)(x+2) ): ( x^2 = A(x+2) + B(x1) )

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

Equate the coefficients of like terms: ( A + B = 0 ) (coefficients of x) ( 2A  B = 1 ) (constant terms)

Solve the system of equations: From equation (1), ( B = A ) Substitute into equation (2): ( 2A  (A) = 1 ) ( 3A = 1 ) ( A = \frac{1}{3} ) ( B = \frac{1}{3} )

Rewrite the original fraction in terms of the partial fractions: ( \frac{x^2}{(x1)(x+2)} = \frac{1/3}{x1}  \frac{1/3}{x+2} )

Integrate each term separately: ( \int \frac{1/3}{x1} , dx = \frac{1}{3} \lnx1 + C_1 ) ( \int \frac{1/3}{x+2} , dx = \frac{1}{3} \lnx+2 + C_2 )
Combine the results: [ \int \frac{x^2}{(x1)(x+2)} , dx = \frac{1}{3} \lnx1  \frac{1}{3} \lnx+2 + C ] Where ( C = C_1 + C_2 ).
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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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