Problem 18 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The indefinite integral of \(\frac{2}{x^{2}-2 x}\) dx is \(-ln|x| + ln|x - 2| + C\).

Step 1: Factorize the denominator

The denominator \(x^{2}-2 x\) can be factorized by taking out a common factor of x. This gives us \(x(x - 2)\).

Step 2: Express the fraction as a sum of partial fractions

Now, express the given function as the sum of partial fractions. This can be done as follows: write \(\frac{2}{x(x - 2)}\) as \(\frac{A}{x} + \frac{B}{x - 2}\). To find the values of A and B, clear the fractions by multiplying both sides by the common denominator \(x(x - 2)\) to get \(2 = A(x - 2) + Bx\). This is an identity, and it should hold for all x. We can choose suitable x values to find A and B. If we let x = 0, the equation becomes 2 = -2A, so A = -1. If we let x = 2, the equation becomes 2 = 2B, so B = 1. Thus, the expression becomes \(-\frac{1}{x} + \frac{1}{x - 2}\).

Step 3: Integrate term-by-term

Now, integrate the terms separately. Remember that the integral of \(\frac{1}{x}\) is \(ln|x|\) and the integral of \(dx\) is x. Then, \(\int \frac{2}{x^{2}-2 x} d x\) becomes \(\int -\frac{1}{x} dx + \int \frac{1}{x - 2} dx\), which evaluates to \(-ln|x| + ln|x - 2| + C\). The C is the constant of integration.