JEE Mains · Maths · STD 12 - 7.1 indefinite integral
For \(I(x)=\int \frac{\sec ^{2} x-2022}{\sin ^{2022} x} d x\), if \(I\left(\frac{\pi}{4}\right)=2^{1011}\), then
- A \(3^{1010} I \left(\frac{\pi}{3}\right)- I \left(\frac{\pi}{6}\right)=0\)
- B \(3^{1010} I \left(\frac{\pi}{6}\right)- I \left(\frac{\pi}{3}\right)=0\)
- C \(3^{1011} I \left(\frac{\pi}{3}\right)- I \left(\frac{\pi}{6}\right)=0\)
- D \(3^{1011} I \left(\frac{\pi}{6}\right)- I \left(\frac{\pi}{3}\right)=0\)
Answer & Solution
Correct Answer
(A) \(3^{1010} I \left(\frac{\pi}{3}\right)- I \left(\frac{\pi}{6}\right)=0\)
Step-by-step Solution
Detailed explanation
\(I(x)=\int \sec ^{2} x \cdot \sin ^{-2022} x d x-2022 \int \sin ^{-2022} x d x\) \(=\tan x \cdot(\sin x)^{-2022}+\int(2022) \tan x \cdot(\sin x)^{-2023} \cos x d x\) \(-2022 \int(\sin x)^{-2022} d x\) \(I ( x )=(\tan x )(\sin x )^{-2022}+ C\) At…
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