Question

Calculus 2 Master needed, show steps with use of partial fraction decomposition

A. ∫(8x^2 - 11x + 6) / (x^3 - 3x^2) dx = -3ln|x| - 5x + C
B. ∫(8x^2 - 11x + 6) / (x^3 - 3x^2) dx = 3ln|x| + 5x + C
C. ∫(8x^2 - 11x + 6) / (x^3 - 3x^2) dx = -2ln|x| - 3x + C
D. ∫(8x^2 - 11x + 6) / (x^3 - 3x^2) dx = 2ln|x| + 3x + C

Answer 1

To solve the integral \( \int(8x^2 - 11x + 6) / (x^3 - 3x^2) dx \), perform partial fraction decomposition, solve for the coefficients, and integrate term by term. Combine and simplify the resulting expressions to find the correct answer.

Step-by-step explanation:

To determine which answer choice is correct for the integral \( \int(8x^2 - 11x + 6) / (x^3 - 3x^2) dx \), we need to perform partial fraction decomposition on the integrand.

First, factor the denominator:

\( x^3 - 3x^2 = x^2(x - 3) \)

Now, set up the partial fractions:

\( \frac{8x^2 - 11x + 6}{x^2(x - 3)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 3} \)

Multiplying through by the common denominator \( x^2(x - 3) \) and equating coefficients, we get a system of equations to solve for A, B, and C. These are:

• \( A(x - 3) + Bx + Cx^2 = 8x^2 - 11x + 6 \)

Matching coefficients of like terms will give us the values of A, B, and C.

After finding A, B, and C, we can integrate term by term:

• \( \int \frac{A}{x} dx = A\ln|x| \)
• \( \int \frac{B}{x^2} dx = -\frac{B}{x} \)
• \( \int \frac{C}{x - 3} dx = C\ln|x - 3| \)

Combining these results and simplifying will lead to either option A, B, C, or D.