Question 9 - CBSE Class 12 Sample Paper for 2024 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at August 14, 2026 by Teachoo

For any integer n, the value of ∫_(-π)^π e^(cos^2 x) sin^3 (2n+1)x dx is

(a) -1 (b) 0 (c) 1 (d) 2

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[MCQ] For any integer n, the value of ∫ e^(cos^2x) sin^3 (2n+1)x dx - CBSE Class 12 Sample Paper for 2024 Boards

part 2 - Question 9 - CBSE Class 12 Sample Paper for 2024 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards - Class 12

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Transcript

This is of the form ∫_(−𝑎)^𝑎▒𝑓(𝑥)𝑑𝑥 where 𝒇(𝒙)=𝒆^(〖𝒄𝒐𝒔〗^𝟐 𝒙) 〖𝐬𝒊𝒏〗^𝟑 (𝟐𝒏+𝟏)𝒙 And, 𝑓(−𝑥)=𝑒^〖𝒄𝒐𝒔〗^𝟐⁡(−𝒙) sin^3⁡〖[(2𝑛+1)〗 (−𝒙)] Since cos (-x) = cos x 𝑓(−𝑥)=𝑒^(〖𝑐𝑜𝑠〗^2⁡𝑥) 〖s𝑖𝑛〗^3⁡[−(2𝑛+1)𝑥] Since sin (-x) = −sin x 𝒇(−𝒙)=〖−𝒆〗^(〖𝒄𝒐𝒔〗^𝟐⁡𝒙) 〖𝒔𝒊𝒏〗^𝟑⁡(𝟐𝒏+𝟏)𝒙 Thus, 𝑓(−𝑥)=−𝑓(𝑥) ∴ ∫130_(−𝝅)^𝝅▒ 𝒆^(〖𝒄𝒐𝒔〗^𝟐 𝒙) 〖𝒔𝒊𝒏〗^𝟑 (𝟐𝒏+𝟏)𝒙𝒅𝒙=𝟎 So, the correct answer is (b)

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