Question

Evaluate ? (8x² + 6x - 1) / (2x³ + 3x² - 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 2x³ + 3x² - 2x = x(2x² + 3x - 2) = x(2x - 1)(x + 2). Since the denominator has three distinct linear factors, the partial fraction decomposition of the integrand has the following form [see this case]. (8x² + 6x - 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 least common denominator, x(2x - 1)(x + 2), obtaining 8x² + 6x - 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 8x² + 6x - 1 = (2A + B + 2C)x² + (3A + 2B - C)x - 2A. The polynomials on each side of the equation above are identical, so the coefficients of corresponding terms must be equal. The coefficient of x² on the right side, 2A + B + 2C, must equal the coefficient of x² 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 = 6 -2A = -1 Solving, we get A = 1/2, B = , and C = , and so we have the following. (Remember to use absolute values where appropriate.) ? (8x² + 6x - 1) / (2x³ + 3x² - 2x) dx = ? [ (1/2)(1/x) + ( ) (1/(2x - 1)) + ( ) (1/(x + 2)) ] dx = + 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.

          Evaluate ? (8x² + 6x - 1) / (2x³ + 3x² - 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
2x³ + 3x² - 2x = x(2x² + 3x - 2) = x(2x - 1)(x + 2).
Since the denominator has three distinct linear factors, the partial fraction decomposition of the integrand has the following form [see this case].
(8x² + 6x - 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 least common denominator, x(2x - 1)(x + 2), obtaining
8x² + 6x - 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
8x² + 6x - 1 = (2A + B + 2C)x² + (3A + 2B - C)x - 2A.
The polynomials on each side of the equation above are identical, so the coefficients of corresponding terms must be equal. The coefficient of x² on the right side, 2A + B + 2C, must equal the coefficient of x² 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 = 6
-2A = -1
Solving, we get A = 1/2, B = , and C = , and so we have the following. (Remember to use absolute values where appropriate.)
? (8x² + 6x - 1) / (2x³ + 3x² - 2x) dx = ? [ (1/2)(1/x) + ( ) (1/(2x - 1)) + ( ) (1/(x + 2)) ] dx
= + 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.

$Evaluate ? (8x² + 6x - 1) / (2x³ + 3x² - 2x) dx. Solution Since the degree of the numerator is less than the degree of the denominator, we don't need to divide. We factor the denominator as 2x³ + 3x² - 2x = x(2x² + 3x - 2) = x(2x - 1)(x + 2). Since the denominator has three distinct linear factors, the partial fraction decomposition of the integrand has the following form [see this case]. (8x² + 6x - 1) / (x(2x - 1)(x + 2)) = A/x + B/(2x - 1) + C/(x + 2) To determine the values of A, B, and C, we multiply both sides of this equation by the least common denominator, x(2x - 1)(x + 2), obtaining 8x² + 6x - 1 = A(2x - 1)(x + 2) + Bx(x + 2) + Cx(2x - 1). Expanding the right side of the equation above and writing it in the standard form for polynomials, we get 8x² + 6x - 1 = (2A + B + 2C)x² + (3A + 2B - C)x - 2A. The polynomials on each side of the equation above are identical, so the coefficients of corresponding terms must be equal. The coefficient of x² on the right side, 2A + B + 2C, must equal the coefficient of x² on the left side—namely, 8. Likewise, the coefficients of x are equal and the constant terms are equal. This gives the following system of equations for A, B, and C. 2A + B + 2C = 8 3A + 2B - C = 6 -2A = -1 Solving, we get A = 1/2, B = , and C = , and so we have the following. (Remember to use absolute values where appropriate.) ? (8x² + 6x - 1) / (2x³ + 3x² - 2x) dx = ? [ (1/2)(1/x) + ( ) (1/(2x - 1)) + ( ) (1/(x + 2)) ] dx = + K In integrating the middle term we have made the mental substitution u = 2x - 1, which gives du = 2 dx and dx = 1/2 du.$

Added by Christopher Y.

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