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

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

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

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

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

JEE Main · 2022 · 29 Jul 2022 · Shift 2 · Indefinite Integrals

For

I (x) = \int \frac{s e c ^{2} x - 2022}{s i n ^{2022} x} dx

, if

I (\frac{π}{4}) = 2^{1011}

, then

Answer: (A)

Using integration by parts,

I (x) = \frac{t a n x}{( s i n x ) ^{2022}} + C

. From the condition

I (π /4) = 2^{1011}

, we find

C = 0

. Thus

I (x) = \frac{t a n x}{( s i n x ) ^{2022}}

. Computing:

I (π /6) = \frac{1/ 3}{( 1/2 ) ^{2022}} = \frac{2 ^{2022}}{3}

and

I (π /3) = \frac{3}{( 3 /2 ) ^{2022}} = \frac{2 ^{2022}}{( 3 ) ^{2021}} = \frac{1}{3 ^{1010}} I (π /6)

. Therefore

3^{1010} I (π /3) = I (π /6)

, or

3^{1010} I (π /3) - I (π /6) = 0

JEE Main · 2022 · 29 Jul 2022 · Shift 2 · Single correct option