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EXAMPLE 2 Evaluate ? (8x^2 + 4x - 1) / (2x^3 + 3x^2 - 2x) dx. SOLUTION Since the degree of the numerator is less than the degree of the denominator, we don't need to divide. We factor the denominator as 2x^3 + 3x^2 - 2x = x(2x^2 + 3x - 2) = x(2x - 1)(x + 2). Since the denominator has three distinct linear factors, the partial fraction decomposition of the integrand has the form (8x^2 + 4x - 1) / (x(2x - 1)(x + 2)) = A/x + B/(2x - 1) + C/(x + 2). To determine the values of A, B, and C, we multiply both sides of this equation by the product of the denominators, x(2x - 1)(x + 2), obtaining 8x^2 + 4x - 1 = A(2x - 1)(x + 2) + Bx(x + 2) + Cx(2x - 1). Expanding the right side of the equation above and writing it in the standard form for polynomials, we get 8x^2 + 4x - 1 = (2A + B + 2C)x^2 + (3A + 2B - C)x - 2A. The polynomials in the equation above are identical, so their coefficients must be equal. The coefficient of x^2 on the right side, 2A + B + 2C, must equal the coefficient of x^2 on the left side--namely, 8. Likewise, the coefficients of x are equal and the constant terms are equal. This gives the following system of equations for A, B, and C. 2A + B + 2C = 8 3A + 2B - C = 4 -2A = -1 Solving, we get A = 1/2, B = 14/5, and C = 21/10, and so we have the following. (Remember to use absolute values where appropriate.) ? (8x^2 + 4x - 1) / (2x^3 + 3x^2 - 2x) dx = ? [1/2 * 1/x + (14/5) * 1/(2x - 1) + (21/10) * 1/(x + 2)] dx = 1/2 ln|x| + 7/5 ln|2x - 1| + 21/10 ln|x + 2| + K In integrating the middle term we have made the mental substitution u = 2x - 1, which gives du = 2 dx and dx = 1/2 du.

          EXAMPLE 2 Evaluate ? (8x^2 + 4x - 1) / (2x^3 + 3x^2 - 2x) dx.

SOLUTION Since the degree of the numerator is less than the degree of the denominator, we don't need to divide. We factor the denominator as
2x^3 + 3x^2 - 2x = x(2x^2 + 3x - 2) = x(2x - 1)(x + 2).
Since the denominator has three distinct linear factors, the partial fraction decomposition of the integrand has the form
(8x^2 + 4x - 1) / (x(2x - 1)(x + 2)) = A/x + B/(2x - 1) + C/(x + 2).
To determine the values of A, B, and C, we multiply both sides of this equation by the product of the denominators, x(2x - 1)(x + 2), obtaining
8x^2 + 4x - 1 = A(2x - 1)(x + 2) + Bx(x + 2) + Cx(2x - 1).
Expanding the right side of the equation above and writing it in the standard form for polynomials, we get
8x^2 + 4x - 1 = (2A + B + 2C)x^2 + (3A + 2B - C)x - 2A.
The polynomials in the equation above are identical, so their coefficients must be equal. The coefficient of x^2 on the right side, 2A + B + 2C, must equal the coefficient of x^2 on the left side--namely, 8. Likewise, the coefficients of x are equal and the constant terms are equal. This gives the following system of equations for A, B, and C.
2A + B + 2C = 8
3A + 2B - C = 4
-2A = -1
Solving, we get A = 1/2, B = 14/5, and C = 21/10, and so we have the following. (Remember to use absolute values where appropriate.)
? (8x^2 + 4x - 1) / (2x^3 + 3x^2 - 2x) dx = ? [1/2 * 1/x + (14/5) * 1/(2x - 1) + (21/10) * 1/(x + 2)] dx
= 1/2 ln|x| + 7/5 ln|2x - 1| + 21/10 ln|x + 2| + K
In integrating the middle term we have made the mental substitution u = 2x - 1, which gives du = 2 dx and dx = 1/2 du.

$EXAMPLE 2 Evaluate ? (8x^2 + 4x - 1) / (2x^3 + 3x^2 - 2x) dx. SOLUTION Since the degree of the numerator is less than the degree of the denominator, we don't need to divide. We factor the denominator as 2x^3 + 3x^2 - 2x = x(2x^2 + 3x - 2) = x(2x - 1)(x + 2). Since the denominator has three distinct linear factors, the partial fraction decomposition of the integrand has the form (8x^2 + 4x - 1) / (x(2x - 1)(x + 2)) = A/x + B/(2x - 1) + C/(x + 2). To determine the values of A, B, and C, we multiply both sides of this equation by the product of the denominators, x(2x - 1)(x + 2), obtaining 8x^2 + 4x - 1 = A(2x - 1)(x + 2) + Bx(x + 2) + Cx(2x - 1). Expanding the right side of the equation above and writing it in the standard form for polynomials, we get 8x^2 + 4x - 1 = (2A + B + 2C)x^2 + (3A + 2B - C)x - 2A. The polynomials in the equation above are identical, so their coefficients must be equal. The coefficient of x^2 on the right side, 2A + B + 2C, must equal the coefficient of x^2 on the left side–namely, 8. Likewise, the coefficients of x are equal and the constant terms are equal. This gives the following system of equations for A, B, and C. 2A + B + 2C = 8 3A + 2B - C = 4 -2A = -1 Solving, we get A = 1/2, B = 14/5, and C = 21/10, and so we have the following. (Remember to use absolute values where appropriate.) ? (8x^2 + 4x - 1) / (2x^3 + 3x^2 - 2x) dx = ? [1/2 * 1/x + (14/5) * 1/(2x - 1) + (21/10) * 1/(x + 2)] dx = 1/2 ln|x| + 7/5 ln|2x - 1| + 21/10 ln|x + 2| + K In integrating the middle term we have made the mental substitution u = 2x - 1, which gives du = 2 dx and dx = 1/2 du.$

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