Question

Evaluate the integral: \[ \int \frac{x+1}{x^3 - 1}dx \]

• A. \( \frac{1}{3} \log \frac{x+1}{x^2 + x + 1} + C \)
• B. \( \frac{1}{3} \log \frac{(x-1)^2}{x^2 + x + 1} + C \)
• C. \( \frac{1}{3} \log \frac{x-1}{x^2 + x + 1} + C \)
• D. \( \frac{1}{3} \log \frac{(x+1)^2}{x^2 - x + 1} + C \)

Hint:

For integrals involving factored denominators, use partial fraction decomposition to separate terms before integrating.

Answer 1

The Correct Option is C

Factorizing the denominator: \[ x^3 - 1 = (x - 1)(x^2 + x + 1) \] Rewriting the fraction: \[ \frac{x+1}{(x-1)(x^2 + x + 1)} \] Using partial fractions: \[ \frac{x+1}{(x-1)(x^2 + x + 1)} = \frac{A}{x-1} + \frac{Bx + C}{x^2 + x + 1} \] Solving for coefficients \( A, B, C \), simplifying, and integrating: \[ \int \frac{x+1}{x^3 - 1} dx = \frac{1}{3} \log \frac{x-1}{x^2 + x + 1} + C \] Thus, the correct answer is: \[ \frac{1}{3} \log \frac{x-1}{x^2 + x + 1} + C \]