perform the partial fraction decomposition of Integrate (2x ^ 2 + 3)/(x * (x - 1) ^ 2) dx

To perform the partial fraction decomposition of the integral ∫(2x^2 + 3)/(x*(x - 1)^2) dx, we first express the integrand as a sum of simpler fractions:

(2x^2 + 3)/(x*(x - 1)^2) = A/x + B/(x - 1) + C/(x - 1)^2

We then multiply through by the common denominator to clear the fractions:

2x^2 + 3 = A*(x - 1)^2 + Bx(x - 1) + C*x

We can then equate coefficients for the powers of x on both sides of the equation to solve for A, B, and C.

For x^2: 2 = A + B For x^1: 0 = -2A + B + C For x^0: 3 = A

From x^0, we know A = 3. Substituting A into the equations for x^2 and x^1, we get:

2 = 3 + B => B = -1 0 = -2*3 -1 + C => C = 5

So, the partial fraction decomposition is:

∫(2x^2 + 3)/(x*(x - 1)^2) dx = ∫(3/x - 1/(x - 1) + 5/(x - 1)^2) dx

Now, you can integrate each term separately.