Question

Evaluate the integral \[ \int \frac{x^4 + 1}{x^6 + 1} dx. \]

• A. \( \tan^{-1} x - \tan^{-1} x^3 + c \)
• B. \( \tan^{-1} x - \frac{1}{3} \tan^{-1} x^3 + c \)
• C. \( \tan^{-1} x + \tan^{-1} x^3 + c \)
• D. \( \tan^{-1} x + \frac{1}{3} \tan^{-1} x^3 + c \)

Hint:

For rational functions, factor and use partial fraction decomposition.

Answer 1

The Correct Option is D

Step 1: Splitting the fraction \[ I = \int \frac{x^4 + 1}{x^6 + 1} dx. \] Factoring: \[ x^6 + 1 = (x^2 + 1)(x^4 - x^2 + 1). \] Step 2: Using partial fractions \[ \frac{x^4 + 1}{x^6 + 1} = \frac{A}{x^2 + 1} + \frac{B}{x^4 - x^2 + 1}. \] Solving for coefficients and integrating leads to: \[ I = \tan^{-1} x + \frac{1}{3} \tan^{-1} x^3 + c. \]