Question

The integral \( \int \frac{1 + x^2 + x^4}{(1 - x^3)(1 + x^3)} \, dx \) is equal to:

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

Hint:

Use partial fraction decomposition when dealing with rational expressions for easier integration.

Answer 1

The Correct Option is C

Step 1: Simplify the integral: \[ \int \frac{1 + x^2 + x^4}{(1 - x^3)(1 + x^3)} \, dx. \] This can be solved by partial fraction decomposition. 
Step 2: After simplification, the result is: \[ \frac{1}{2} \log(1+x) - \log(1-x) + C. \]