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. $3^{1011} \cdot I\left(\frac{\pi}{6}\right) - I\left(\frac{\pi}{3}\right) = 0$

Answer 1

Answer: (A)

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

Answer 2

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

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

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

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

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

$⇒c=2^{1011}−2^{1011}$
$⇒c=0$

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

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