Question

The form of the partial fraction decomposition of a rational function is given below.

4x²+3x+32/x-4(x²+9) =A/x-4+Bx+c/x²+9.
A= , B=, C= ,Now evaluate the indefinite integral. ∫ 4x²+3x+32/x-4(x²+9)dx= Note You can earn partial credit on this probiem

Answer 1

The question is about determining the coefficients for a partial fraction decomposition of a rational function and integrating that function. The process involves equating coefficients to find the constants and then using standard integration techniques.

Step-by-step explanation:

The student's question involves finding the constants A, B, and C for the partial fraction decomposition of the rational function 4x²+3x+32 over (x-4)(x²+9), and then evaluating the indefinite integral of that function.

To find A, B, and C, we set up the equation 4x²+3x+32 = A(x²+9) + (Bx+C)(x-4) and solve for the coefficients by comparing the coefficients of x², x, and the constant terms on both sides of the equation.

Once we have the values of A, B, and C, we can integrate the decomposed function by integrating each part separately, which involves straightforward polynomial integration and the integral of the form ∫ dx/(x²+a²),

which is a standard integral resulting in a tan⁻¹(x/a).