How do you integrate #int (2x+1) /( (x2)(x^2+4)# using partial fractions?
Let:
The integrand can be broken down into a partial fraction as follows:
Resulting in:
Comparing Coefficients:
And so:
Thus, what we get from the partial fraction decomposition is:
We can simply evaluate the first integral directly.
By dividing the second integral into two fractions, we can handle it:
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To integrate ( \int \frac{2x + 1}{(x  2)(x^2 + 4)} ) using partial fractions, follow these steps:
 Factor the denominator into irreducible factors.
 Express the fraction as a sum of partial fractions with undetermined coefficients.
 Find the values of the undetermined coefficients.
 Integrate each partial fraction.
 Combine the results.
Given ( \int \frac{2x + 1}{(x  2)(x^2 + 4)} ):

Factor the denominator: ( (x  2)(x^2 + 4) = (x  2)(x + 2i)(x  2i) ).

Express the fraction as partial fractions: [ \frac{2x + 1}{(x  2)(x^2 + 4)} = \frac{A}{x  2} + \frac{Bx + C}{x^2 + 4} ]

Multiply both sides by the denominator to clear the fractions: [ 2x + 1 = A(x^2 + 4) + (Bx + C)(x  2) ]

Expand and equate coefficients: [ 2x + 1 = Ax^2 + 4A + Bx^2  2Bx + Cx  2C ]

Group like terms: [ 2x + 1 = (A + B)x^2 + (2B + C)x + (4A  2C) ]

Equate coefficients:
 For ( x^2 ): ( A + B = 0 ) (1)
 For ( x ): ( 2B + C = 2 ) (2)
 For constant term: ( 4A  2C = 1 ) (3)

Solve the system of equations (1), (2), and (3) for ( A ), ( B ), and ( C ).

Once you find the values of ( A ), ( B ), and ( C ), integrate each term separately.

Finally, combine the results and simplify the expression if necessary.
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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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