What is the integral of x (1 - x⁴) dx = ?

• A: (1/4) log |1 + x²| + C
• B: (1/4) log |1 - x²| + C
• C: (1/2) log |1 + x²| + C
• D: None of these

To find the integral of the function \( x(1 - x^4) \), we can break it down into simpler parts. The integral can be expressed as:

Step-by-Step Integration

We start with:

∫ x(1 - x^4) dx

Distributing the Terms

This can be rewritten as:

∫ (x - x^5) dx

Integrating Each Term

• For ∫ x dx, the result is (1/2)x².
• For ∫ x^5 dx, the result is (1/6)x^6.

Combining the Results

Putting it all together, we have:

(1/2)x² - (1/6)x^6 + C

Final Answer

This does not match any of the provided options A, B, or C. Therefore, the correct choice is:

D: None of these