Question

For \(I(x) = \int \frac{\sec^2 x - 2022}{\sin^{2022} x} \, dx, \quad \text{if } I\left(\frac{\pi}{4}\right) = 2^{1011}\), then

• A. \(3^{1010} \cdot I\left(\frac{\pi}{3}\right) - I\left(\frac{\pi}{6}\right) = 0\)
• B. \(3^{1010} \cdot I\left(\frac{\pi}{6}\right) - I\left(\frac{\pi}{3}\right) = 0\)
• C. \(3^{1011} \cdot I\left(\frac{\pi}{3}\right) - I\left(\frac{\pi}{6}\right) = 0\)
• D. \(3^{1011} \cdot I\left(\frac{\pi}{6}\right) - I\left(\frac{\pi}{3}\right) = 0\)

Hint:

\(I(x) = \int \frac{\sec^2 x - 2022}{\sin^{2022} x} \, dx\)

\(= \int (\sec^2 x \cdot \sin^{-2022} x - 2022 \sin^{-2022} x) \, dx\)

\(= \sin^{-2022} x \tan x + \int 2022 \sin^{-2023} x \cos x \tan x \, dx - \int 2022 \sin^{-2022} x \, dx + C\)

\(I(x) = \sin^{-2022} x \tan x + c\)

\(\therefore I\left(\frac{\pi}{4}\right) = 2^{1011}\)

\(⇒c=2^{1011}−2^{1011}\)
\(⇒c=0\)

\(I(\frac{\pi}{3}) = (\frac{2}{\sqrt{3}} )^{2022}\sqrt{3}\)

\(I\left(\frac{\pi}{6}\right) = 2^{20221}\frac{1}{\sqrt{3}}\)
So, option (A): \(3^{1010} \cdot I\left(\frac{\pi}{3}\right) - I\left(\frac{\pi}{6}\right) = 0\)
 

Answer 1

The Correct Option is A

To solve the given integral problem, we start by analyzing the function:

\(I(x) = \int \frac{\sec^2 x - 2022}{\sin^{2022} x} \, dx\)

We are given a condition:

\( I\left(\frac{\pi}{4}\right) = 2^{1011}\)

Our goal is to find which of the following identities holds true:

1. \(3^{1010} \cdot I\left(\frac{\pi}{3}\right) - I\left(\frac{\pi}{6}\right) = 0\)
2. \(3^{1010} \cdot I\left(\frac{\pi}{6}\right) - I\left(\frac{\pi}{3}\right) = 0\)
3. \(3^{1011} \cdot I\left(\frac{\pi}{3}\right) - I\left(\frac{\pi}{6}\right) = 0\)
4. \(3^{1011} \cdot I\left(\frac{\pi}{6}\right) - I\left(\frac{\pi}{3}\right) = 0\)

Let's evaluate the integral by considering its symmetry properties and any potential substitutions or transformations. Notice that the trigonometric identities and powers will help simplify the expressions at specific angles like \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{6}\).

Using trigonometric identities:

- At \(x = \frac{\pi}{4}\), we know \(\sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\) and \(\sec^2\left(\frac{\pi}{4}\right) = 2\).

- We use symmetry and properties of trigonometric functions to evaluate expressions at \(\frac{\pi}{3}\) and \(\frac{\pi}{6}\).

Next, compute these values:

\(I\left(\frac{\pi}{3}\right)\) and \(I\left(\frac{\pi}{6}\right)\)

By symmetry and trigonometric identities similar to those at \(\frac{\pi}{4}\), we can deduce the relationship between these values that satisfies the condition:

The identity that satisfies this condition is:

\(3^{1010} \cdot I\left(\frac{\pi}{3}\right) - I\left(\frac{\pi}{6}\right) = 0\)

Thus, the correct answer is option 1.