Key Concepts and Formulas

• Integration by parts: $\int u \, dv = uv - \int v \, du$
• Trigonometric identities: $\sec x = \frac{1}{\cos x}$, $\tan x = \frac{\sin x}{\cos x}$
• Power rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$

Step-by-Step Solution

Step 1: Rewrite the integral and prepare for integration by parts. We are given the integral $I(x) = \int \frac{\sec^2 x - 2022}{\sin^{2022} x} dx = \int \frac{\sec^2 x}{\sin^{2022} x} dx - \int \frac{2022}{\sin^{2022} x} dx$ We aim to use integration by parts on the first integral.

Step 2: Apply integration by parts to the first integral. Let $u = \frac{1}{\sin^{2022} x}$ and $dv = \sec^2 x \, dx$. Then $du = \frac{-2022 \cos x}{\sin^{2023} x} dx$ and $v = \tan x$. Applying integration by parts, we have $\int \frac{\sec^2 x}{\sin^{2022} x} dx = \frac{\tan x}{\sin^{2022} x} - \int \tan x \cdot \frac{-2022 \cos x}{\sin^{2023} x} dx = \frac{\tan x}{\sin^{2022} x} + \int \frac{2022 \sin x}{\cos x} \cdot \frac{\cos x}{\sin^{2023} x} dx$ $= \frac{\tan x}{\sin^{2022} x} + \int \frac{2022}{\sin^{2022} x} dx$

Step 3: Substitute the result back into the original integral. Substituting this back into the original integral, we get $I(x) = \frac{\tan x}{\sin^{2022} x} + \int \frac{2022}{\sin^{2022} x} dx - \int \frac{2022}{\sin^{2022} x} dx = \frac{\tan x}{\sin^{2022} x} + C$

Step 4: Determine the constant of integration using the given condition. We are given that $I\left(\frac{\pi}{4}\right) = 2^{1011}$. Thus, $2^{1011} = \frac{\tan(\pi/4)}{\sin^{2022}(\pi/4)} + C = \frac{1}{(1/\sqrt{2})^{2022}} + C = (\sqrt{2})^{2022} + C = (2^{1/2})^{2022} + C = 2^{1011} + C$ Therefore, $C = 0$.

Step 5: Write the final expression for $I(x)$. $I(x) = \frac{\tan x}{\sin^{2022} x}$

Step 6: Evaluate $I(\pi/3)$ and $I(\pi/6)$. $I\left(\frac{\pi}{3}\right) = \frac{\tan(\pi/3)}{\sin^{2022}(\pi/3)} = \frac{\sqrt{3}}{(\sqrt{3}/2)^{2022}} = \frac{\sqrt{3}}{(\sqrt{3})^{2022}/2^{2022}} = \frac{\sqrt{3} \cdot 2^{2022}}{3^{1011}}$ $I\left(\frac{\pi}{6}\right) = \frac{\tan(\pi/6)}{\sin^{2022}(\pi/6)} = \frac{1/\sqrt{3}}{(1/2)^{2022}} = \frac{2^{2022}}{\sqrt{3}}$

Step 7: Check option (A). $3^{1010} I\left(\frac{\pi}{3}\right) - I\left(\frac{\pi}{6}\right) = 3^{1010} \cdot \frac{\sqrt{3} \cdot 2^{2022}}{3^{1011}} - \frac{2^{2022}}{\sqrt{3}} = \frac{\sqrt{3} \cdot 2^{2022}}{3} \cdot 3^{1010} - \frac{2^{2022}}{\sqrt{3}} = \frac{2^{2022}}{\sqrt{3}} - \frac{2^{2022}}{\sqrt{3}} = 0$ Thus, option (A) is correct.

Step 8: Briefly check option (B). $3^{1010} I\left(\frac{\pi}{6}\right) - I\left(\frac{\pi}{3}\right) = 3^{1010} \cdot \frac{2^{2022}}{\sqrt{3}} - \frac{\sqrt{3} \cdot 2^{2022}}{3^{1011}} = \frac{3^{1010} \cdot 2^{2022}}{\sqrt{3}} - \frac{2^{2022}}{\sqrt{3}} \neq 0$

Common Mistakes & Tips

• Remember to include the constant of integration, $C$, when evaluating indefinite integrals.
• Be careful with trigonometric identities and simplifying expressions.
• Double-check the integration by parts steps to avoid errors in the derivative and integral.

Summary

We first evaluated the indefinite integral $I(x)$ using integration by parts. We then used the given condition $I(\pi/4) = 2^{1011}$ to find the constant of integration. Finally, we evaluated $I(\pi/3)$ and $I(\pi/6)$ and substituted them into the options to find the correct answer.

Final Answer

The final answer is \boxed{0}, which corresponds to option (A).