1. Problem: Calculate the indefinite integral $$\int \frac{7x - 2}{x^2 - 1} dx$$. 2. Formula and rules: Use partial fraction decomposition since the denominator factors as $$x^2 - 1 = (x-1)(x+1)$$. 3. Decompose: $$\frac{7x - 2}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1}$$ Multiply both sides by $$x^2 - 1$$: $$7x - 2 = A(x+1) + B(x-1)$$ 4. Expand and group terms: $$7x - 2 = A x + A + B x - B = (A + B) x + (A - B)$$ 5. Equate coefficients: For $$x$$: $$7 = A + B$$ For constant: $$-2 = A - B$$ 6. Solve the system: Add equations: $$7 + (-2) = (A + B) + (A - B) \Rightarrow 5 = 2A \Rightarrow A = \frac{5}{2}$$ Substitute $$A$$ into $$7 = A + B$$: $$7 = \frac{5}{2} + B \Rightarrow B = 7 - \frac{5}{2} = \frac{9}{2}$$ 7. Rewrite integral: $$\int \frac{7x - 2}{x^2 - 1} dx = \int \frac{5/2}{x-1} dx + \int \frac{9/2}{x+1} dx$$ 8. Integrate: $$= \frac{5}{2} \int \frac{1}{x-1} dx + \frac{9}{2} \int \frac{1}{x+1} dx = \frac{5}{2} \ln|x-1| + \frac{9}{2} \ln|x+1| + C$$ Final answer: $$\int \frac{7x - 2}{x^2 - 1} dx = \frac{5}{2} \ln|x-1| + \frac{9}{2} \ln|x+1| + C$$