JEE Main 2022 · 29 Jul · Shift 2 · Mathematics Q8

← JEE Main 2022 · 29 Jul · Shift 2

MathematicsSingle correct+4 / -1Question 8 of 90

For $I(x)=\displaystyle\int\frac{\sec^2x-2022}{\sin^{2022}x}\,dx$ , if $I\left(\dfrac\pi4\right)=2^{1011}$ , then

3^{1010}I\left(\dfrac\pi3\right)-I\left(\dfrac\pi6\right)=0

3^{1010}I\left(\dfrac\pi6\right)-I\left(\dfrac\pi3\right)=0

3^{1011}I\left(\dfrac\pi3\right)-I\left(\dfrac\pi6\right)=0

3^{1011}I\left(\dfrac\pi6\right)-I\left(\dfrac\pi3\right)=0

Answer and worked solution

Official answer: A

Spot the derivative. $\dfrac d{dx}\left(\dfrac{\tan x}{\sin^{2022}x}\right)=\dfrac{\sec^2x}{\sin^{2022}x}-\dfrac{2022}{\sin^{2022}x}$ , so $I=\dfrac{\tan x}{\sin^{2022}x}+C$ , and $I\left(\frac\pi4\right)=2^{1011}$ gives $C=0$ .

Values. $I\left(\frac\pi3\right)=\dfrac{\sqrt3\,2^{2022}}{3^{1011}}$ , $I\left(\frac\pi6\right)=\dfrac{2^{2022}}{\sqrt3}=3^{1010}I\left(\frac\pi3\right)$ .

(A)

Topics:Standard integrals Trigonometric ratios and identities

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JEE Main 2022 · 29 Jul · Shift 2, Mathematics question 8