CBSE Class 12 Sample Paper for 2024 Boards

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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 Dec. 13, 2024 by

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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Transcript

This is of the form ∫_(βˆ’π‘Ž)^π‘Žβ–’π‘“(π‘₯)𝑑π‘₯ where 𝒇(𝒙)=𝒆^(〖𝒄𝒐𝒔〗^𝟐 𝒙) γ€–π¬π’Šπ’γ€—^πŸ‘ (πŸπ’+𝟏)𝒙 And, 𝑓(βˆ’π‘₯)=𝑒^〖𝒄𝒐𝒔〗^𝟐⁑(βˆ’π’™) sin^3⁑〖[(2𝑛+1)γ€— (βˆ’π’™)] Since cos (-x) = cos x 𝑓(βˆ’π‘₯)=𝑒^(γ€–π‘π‘œπ‘ γ€—^2⁑π‘₯) γ€–s𝑖𝑛〗^3⁑[βˆ’(2𝑛+1)π‘₯] Since sin (-x) = βˆ’sin x 𝒇(βˆ’π’™)=γ€–βˆ’π’†γ€—^(〖𝒄𝒐𝒔〗^πŸβ‘π’™) γ€–π’”π’Šπ’γ€—^πŸ‘β‘(πŸπ’+𝟏)𝒙 Thus, 𝑓(βˆ’π‘₯)=βˆ’π‘“(π‘₯) ∴ ∫130_(βˆ’π…)^π…β–’β€Š 𝒆^(〖𝒄𝒐𝒔〗^𝟐 𝒙) γ€–π’”π’Šπ’γ€—^πŸ‘ (πŸπ’+𝟏)𝒙𝒅𝒙=𝟎 So, the correct answer is (b)

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo