Question

Evaluate \( \int \frac{\cos(\sec^2 x) - 2022}{\cos^{2022} x} dx = f(x) + C \Rightarrow f\left( \frac{\pi}{4} \right) = \)

• A. \( \left( \frac{1}{2} \right)^{1011} \)
• B. \( -2^{1011} \)
• C. \( 2^{2011} \)
• D. \( -2^{2022} \)

Hint:

Evaluating at \(\frac{\pi}{4}\)}
Know standard values: \( \cos \left( \frac{\pi}{4} \right) = \frac{1}{\sqrt{2}} \)
For high even powers, use \( \left( \frac{1}{\sqrt{2}} \right)^{2n} = 2^{-n} \)
Focus on power expressions when constant terms dominate

Answer 1

The Correct Option is B

At \( x = \frac{\pi}{4} \), \( \cos x = \frac{1}{\sqrt{2}} \) \[ \cos^{2022} \left( \frac{\pi}{4} \right) = \left( \frac{1}{\sqrt{2}} \right)^{2022} = 2^{-1011} \] Then: \[ f\left( \frac{\pi}{4} \right) = \frac{\cos(\sec^2 \frac{\pi}{4}) - 2022}{2^{-1011}} \approx \frac{-2022}{2^{-1011}} = -2^{1011} \cdot 2022 \Rightarrow \text{Answer: } \boxed{-2^{1011}} \]