Question

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

• A. \( x + \log|x| + \frac{1}{2} \log(x^2 + 1) + \sin^{-1}(x) + c \)
• B. \( x - \log|x| + \frac{1}{2} \log(x^2 + 1) - \sin^{-1}(x) + c \)
• C. \( x + \log|x| - \frac{1}{2} \log(x^2 + 1) + \tan^{-1}(x) + c \)
• D. \( x - \log|x| + \frac{1}{2} \log(x^2 + 1) - \tan^{-1}(x) + c \)

Hint:

Remember to break complex rational expressions into simpler parts for easier integration using partial fraction decomposition.