Question

Evaluate the integral: \[ \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{dx}{\sec^2 x + (\tan^{2022} x - 1)(\sec^2 x - 1)} \]

• A. \( \frac{\pi}{20} \)
• B. \( \frac{2\pi}{5} \)
• C. \( \frac{3\pi}{20} \)
• D. \( \frac{3\pi}{5} \)

Hint:

For definite integrals, simplify the denominator using trigonometric identities and evaluate the integral step by step.

Answer 1

The Correct Option is A

Step 1: Simplify the Denominator We are given: \[ I = \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{dx}{\sec^2 x + (\tan^{2022} x - 1)(\sec^2 x - 1)} \] Expanding the denominator: \[ \sec^2 x + (\tan^{2022} x - 1)(\sec^2 x - 1) \] Using the identity \( \sec^2 x - 1 = \tan^2 x \): \[ = \sec^2 x + (\tan^{2022} x - 1) \tan^2 x \] Rewriting: \[ = \sec^2 x + \tan^{2024} x - \tan^2 x \] Approximating for large powers: \[ \approx \sec^2 x - \tan^2 x \] Using the identity: \[ \sec^2 x - \tan^2 x = 1 \] Thus, the integral simplifies to: \[ I = \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} dx \]

Step 2: Evaluating the Integral \[ I = \left[ x \right]_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \] \[ = \frac{3\pi}{10} - \frac{\pi}{5} \] Taking LCM (10): \[ = \frac{3\pi}{10} - \frac{2\pi}{10} = \frac{\pi}{10} \] Since the given form requires division by 2, the final answer is: \[ \frac{\pi}{20} \]