Question

Evaluate the integral: $ \int \frac{1}{x(x^4 + 1)} \, dx $

• A. \( \frac{1}{2} \log |x^4 + 1| \)
• B. \( \frac{1}{x^4 + 1} \)
• C. \( \frac{1}{2} \log |x| \)
• D. \( \frac{1}{x} \)

Hint:

Partial fraction decomposition is a helpful technique when dealing with rational functions in integrals.

Answer 1

The Correct Option is A

We are asked to evaluate: \[ \int \frac{1}{x(x^4 + 1)} \, dx \] We will use partial fraction decomposition to simplify the integrand. The integrand can be expressed as: \[ \frac{1}{x(x^4 + 1)} = \frac{A}{x} + \frac{Bx + C}{x^4 + 1} \] After performing partial fraction decomposition (details omitted for brevity), we integrate each term. The result of this integral is: \[ \frac{1}{2} \log |x^4 + 1| + C \] 
Thus, the correct answer is \( \frac{1}{2} \log |x^4 + 1| \).