Question

Use partial fractions to find the indefinite integral. $$ \int \frac{x^{2}-4 x-4}{x^{3}-4 x} d x $$

Short answer

The indefinite integral of the given function is \(-2\ln|x| - \ln|x-2| + 2\ln|x+2| + C\).

Step-by-step solution

Perform Polynomial Division

First, we need to check if the degree of the numerator \(x^{2}-4 x-4\) is greater than the degree of the denominator \(x^{3}-4 x\). In this case, the degree of the denominator is greater, so, we don't need to perform polynomial division.

Decompose into Partial Fraction

Next, we can factorize \(x^{3}-4 x\) into \(x(x^{2}-4)\) or \(x(x-2)(x+2)\). Our fraction becomes: \[\frac{x^{2}-4 x-4}{x(x-2)(x+2)}\]. Then we decompose the fraction into partial fractions. We write it in the form \[\frac{A}{x} + \frac{B}{x-2} + \frac{C}{x+2}\] where A, B, and C are constants to be found.

Find Constants A, B, and C

Multiply both sides by the denominator \(x(x-2)(x+2)\) to clear the fractions. Now solve for A, B, and C, we get: \(A = -2, B = -1, C = 2\). Therefore, the fraction decomposed into partial fractions is \[= -\frac{2}{x} - \frac{1}{x-2} + \frac{2}{x+2}\].

Compute Integral

The next step is to integrate our decomposed fraction. By integrating each term separately, the integral becomes \[-2\int\frac{dx}{x} - \int\frac{dx}{x-2} + 2\int\frac{dx}{x+2}\] which simplifies to \[-2\ln|x| - \ln|x-2| + 2\ln|x+2| + C\]. C is the constant of integration.