Question

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

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

Hint:

When faced with complicated rational functions, use partial fractions to break them down into simpler components for easier integration.

Answer 1

The Correct Option is B

We are asked to evaluate the integral: \[ \int \frac{1}{x(x^4 + 1)} \, dx \] Using partial fraction decomposition, we can write the integrand as: \[ \frac{1}{x(x^4 + 1)} = \frac{A}{x} + \frac{B x^3 + C x^2 + D x + E}{x^4 + 1} \] Simplifying and solving the coefficients (details omitted for brevity), we get: \[ \int \frac{1}{x(x^4 + 1)} \, dx = \frac{1}{2} \log |x^4 + 1| \]
Thus, the correct answer is \( \frac{1{2} \log |x^4 + 1|} \).