Since the denominator has three distinct linear factors, the partial fraction decomposition of the integrand has the form†8x2 + 9x − 1x(2x − 1)(x + 2) =

The partial fraction decomposition of the given function is done by expressing the function as a sum of fractions where each fraction has a simpler denominator.

The given function is:

f(x) = (8x^2 + 9x - 1) / (x * (2x - 1) * (x + 2))

The partial fraction decomposition of this function will have the form:

f(x) = A/x + B/(2x - 1) + C/(x + 2)

where A, B, and C are constants that we need to find.

To find these constants, we multiply both sides of the equation by the common denominator x * (2x - 1) * (x + 2) to get rid of the fractions:

8x^2 + 9x - 1 = A * (2x - 1) * (x + 2) + B * x * (x + 2) + C * x * (2x - 1)

Now, we can find the values of A, B, and C by comparing the coefficients on both sides of the equation or by substituting suitable values for x.