Question

Evaluate \( \int_{1}^{2} \frac{x^4 - 1}{x^6 - 1} dx \):

• A. \( 1 \)
• B. \( \frac{121}{6} \)
• C. \( \sqrt{2} -1 \)
• D. \( \frac{1}{\sqrt{3}} \tan^{-1} \left( \frac{\sqrt{3}}{2} \right) \)

Hint:

Factorize the denominator and check for common factors before applying integration techniques.

Answer 1

The Correct Option is A

Step 1: Simplifying the integrand. Rewriting the given integral: \[ I = \int_{1}^{2} \frac{x^4 - 1}{x^6 - 1} dx. \] Factorizing numerator and denominator: \[ x^4 - 1 = (x^2 -1)(x^2 +1), \] \[ x^6 -1 = (x^2 -1)(x^4 + x^2 +1). \] Cancelling the common term \( (x^2 -1) \), we get: \[ I = \int_{1}^{2} \frac{x^2 + 1}{x^4 + x^2 + 1} dx. \] Step 2: Splitting into Partial Fractions. Using substitution \( t = x^2 \) and rewriting the denominator in solvable form: \[ I = \int \frac{dt}{t^2 + t + 1}. \] Solving using trigonometric substitution or completing the square leads to: \[ I = 1. \]