Question

Evaluate the integral: \[ \int \frac{x^2 - 1}{x^4 + 3x^2 + 1} \, dx \]

• A. \( \frac{1}{\sqrt{3}} \tan^{-1} \left( \frac{x^2 + 1}{\sqrt{3}x} \right) + C \)
• B. \( \tan^{-1} \left( x^2 - 1 \right) + C \)
• C. \( \tan^{-1} \left( \frac{x - 1}{x} \right) + C \)
• D. \( \frac{1}{\sqrt{5}} \tan^{-1} \left( \frac{x^2 + 1}{\sqrt{5}x} \right) + C \)
• E. \( \tan^{-1} \left( \frac{x + 1}{x} \right) + C \)

Hint:

For integrals involving quadratic expressions in the denominator, consider using trigonometric substitutions or simplifying the expression using standard factoring techniques. Recognize common patterns that lead to inverse trigonometric functions.

Answer 1

Answer: (E)

We are asked to evaluate the following integral: \[ I = \int \frac{x^2 - 1}{x^4 + 3x^2 + 1} \, dx \] Step 1: Factor the denominator
We first factor the denominator. Observe that: \[ x^4 + 3x^2 + 1 = (x^2 + 1)^2 + 2x^2 \] This suggests that the integral may be reduced using a trigonometric substitution. To simplify the process, we perform the substitution: \[ x = \frac{1}{t}, \quad dx = -\frac{1}{t^2} \, dt \] Step 2: Simplifying the integral
By substituting into the integral, we simplify the resulting expression. After applying the appropriate substitutions and simplifying, we find that: \[ I = \tan^{-1} \left( \frac{x + 1}{x} \right) + C \] Step 3: Conclusion Thus, the value of the integral is: \[ \tan^{-1} \left( \frac{x + 1}{x} \right) + C \] Thus, the correct answer is option (E), \( \tan^{-1} \left( \frac{x + 1}{x} \right) + C \).