Question

\(\displaystyle \int \frac{\sqrt{x^4 + x^{-4} + 2}}{x^3} dx =\)

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

Hint:

Always try to simplify complex expressions under square roots. Often they form perfect squares!

Answer 1

The Correct Option is A

First simplify the integrand: \[ x^4 + x^{-4} + 2 = \left(x^2 + x^{-2}\right)^2 \Rightarrow \sqrt{x^4 + x^{-4} + 2} = x^2 + x^{-2} \] So integrand becomes: \[ \frac{x^2 + x^{-2}}{x^3} = x^{-1} + x^{-5} \] Now integrate: \[ \int \left(x^{-1} + x^{-5} \right) dx = \log |x| - \frac{1}{4x^4} + C \]