How do you integrate #int (x^3 + 2x  1) / (2x^2  3x  2)# using partial fractions?
First, we need to obtain a polynomial in the numerator that is of a lower grade than the one in the denominator.
Through extended division:
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To integrate the function ( \frac{{x^3 + 2x  1}}{{2x^2  3x  2}} ) using partial fractions, follow these steps:

Factor the denominator (2x^2  3x  2) into linear factors. (2x^2  3x  2 = (2x + 1)(x  2))

Write the fraction as a sum of partial fractions with undetermined constants. ( \frac{{x^3 + 2x  1}}{{2x^2  3x  2}} = \frac{A}{2x + 1} + \frac{B}{x  2})

Multiply both sides by the denominator to clear the fractions. (x^3 + 2x  1 = A(x  2) + B(2x + 1))

Expand and equate coefficients of like terms. (x^3 + 2x  1 = Ax  2A + 2Bx + B)

Group like terms and equate coefficients. (x^3 + 2x  1 = (A + 2B)x + (2A + B))

Equate coefficients: (A + 2B = 1) (for the coefficient of (x^3)) (2A + B = 2) (for the constant term)

Solve the system of equations to find the values of (A) and (B).

Once you find the values of (A) and (B), substitute them back into the partial fraction decomposition.

Integrate each term separately.

Finally, sum up the integrals to get the result.
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To integrate ( \int \frac{x^3 + 2x  1}{2x^2  3x  2} ) using partial fractions, first factorize the denominator. It factors as ( (2x + 1)(x  2) ). Then, you decompose the rational function into partial fractions:
[ \frac{x^3 + 2x  1}{(2x + 1)(x  2)} = \frac{A}{2x + 1} + \frac{B}{x  2} ]
Next, you find the values of ( A ) and ( B ) by multiplying both sides by the denominator and comparing coefficients. After finding the values of ( A ) and ( B ), you integrate each term separately.
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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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