91Ó°ÊÓ

First, we need to factor the denominator \(x^3 - 4x\). Notice that both terms share a common factor of \(x\). We can factor this out: $$x^3 - 4x = x(x^2 - 4)$$ Now we also have a difference of squares in the second term \((x^2 - 4)\). We can factor this as well: $$x(x^2 - 4) = x(x + 2)(x - 2)$$ So the factored denominator is \(x(x + 2)(x - 2)\).

We will now decompose the integrand \(\frac{x^2 + 12x - 4}{x(x + 2)(x - 2)}\) into partial fractions. We do this by setting up unknown coefficients for each factor in the denominator: $$\frac{x^2 + 12x - 4}{x(x + 2)(x - 2)} = \frac{A}{x} + \frac{B}{x + 2} + \frac{C}{x - 2}$$ Now we will multiply each side of the equation by the denominator \(x(x + 2)(x - 2)\) to clear the fractions: $$x^2 + 12x - 4 = A(x + 2)(x - 2) + Bx(x - 2) + Cx(x + 2)$$

Now we will solve for the coefficients A, B, and C. There are a few ways to do this, and we will use the method of equating coefficients. We can expand each term and group the x terms together: $$x^2 + 12x - 4 = Ax^2 - 4A + Bx^2 - 2Bx + Cx^2 + 2Cx$$ Combine the x terms: $$x^2 + 12x - 4 = (A + B + C)x^2 + (-2B + 2C)x - 4A$$ Now we can equate the terms: $$A + B + C = 1$$ $$-2B + 2C = 12$$ $$-4A = -4$$ From the last equation, we have \(A = 1\). Plugging this into the first equation, we get: $$1 + B + C = 1 \Rightarrow B + C = 0 \Rightarrow C = -B$$ Now we can substitute this into the second equation: $$-2B + 2(-B) = 12 \Rightarrow -4B = 12$$ From this, we find \(B = -3\). Finally, we can substitute this back into the equation for C: $$C = -(-3) = 3$$ So the coefficients are \(A = 1\), \(B = -3\), and \(C = 3\).

Now that we have the coefficients, we can substitute them back into the decomposed integrand: $$\frac{x^2 + 12x - 4}{x(x + 2)(x - 2)} = \frac{1}{x} - \frac{3}{x + 2} + \frac{3}{x - 2}$$

Now we can integrate each term separately: $$\int \frac{x^2 + 12x - 4}{x(x + 2)(x - 2)} dx = \int \frac{1}{x} dx - 3\int \frac{1}{x + 2} dx + 3 \int \frac{1}{x - 2} dx$$ These are all straightforward integrals: $$= ln|x| - 3ln|x + 2| + 3ln|x - 2| + C$$ We can write the final answer with a single logarithm using properties of logarithms: $$= ln\left|\frac{x(x - 2)^3}{(x + 2)^3}\right| + C$$