JEE Advanced 2022 Paper 2 · Mathematics · Indefinite and Definite Integrals · hard

For a non-negative integer n let In=∫0^π/2sin ^nx dx.

One or more options correct

For a non-negative integer

n

let

I_n=\displaystyle\int_0^{\pi/2}\sin^{n}x\,dx

. Which of the following statements is/are correct?

Show answer and solution

Answer: A

Integration by parts gives

I_n=\frac{n-1}{n}I_{n-2}

, with

I_0=\frac\pi2

and

I_1=1

. Then

I_3=\frac23

and

I_5=\frac45\cdot\frac23=\frac8{15}

; A is true. Also

I_2=\frac\pi4

I_4=\frac{3\pi}{16}

and

I_6=\frac56\cdot\frac{3\pi}{16}=\frac{5\pi}{32}

, so B is false (it quotes

I_4

). From

nI_nI_{n-1}=\frac\pi2

and

I_n\sim I_{n-1}

we get

nI_n^2\to\frac\pi2

, so

I_n\sim\sqrt{\pi/(2n)}

; the series therefore diverges like

\sum n^{-1/2}

, making C false, and

\sqrt n\,I_n\to\sqrt{\pi/2}\neq\frac\pi2

, making D false.

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