Question

What is the correct form of the partial fraction decomposition for the integral:

∫ 14(x⁸ + 4)/((x - 5)(x² - 1)²(x² + 5)²) dx
A. ∫ A/(x - 5) dx + ∫ (Bx + C)/((x² - 1)²) dx + ∫ (Dx + E)/((x² + 5)²) dx

Answer 1

The correct form of the partial fraction decomposition for the given integral is A/(x - 5) + (Bx + C)/((x² - 1)²) + (Dx + E)/((x² + 5)²). The values of the coefficients A, B, C, D, and E can be found by equating numerators and comparing coefficients.

Step-by-step explanation:

The correct form of the partial fraction decomposition for the integral ∫ 14(x⁸ + 4)/((x - 5)(x² - 1)²(x² + 5)²) dx is:

∫ A/(x - 5) dx + ∫ (Bx + C)/((x² - 1)²) dx + ∫ (Dx + E)/((x² + 5)²) dx

To find the values of A, B, C, D, and E, we can first find a common denominator and combine the fractions on the right side. Then, we equate the numerators and solve for the unknown coefficients by comparing coefficients.