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Ex 7.11, 1 By using the properties of definite integrals, evaluate the integrals : ∫_0^(πœ‹/2)β–’γ€–cos^2⁑π‘₯ 𝑑π‘₯γ€— Let 𝐈=∫_𝟎^(𝝅/𝟐)▒〖〖𝒄𝒐𝒔〗^πŸβ‘π’™ 𝒅𝒙〗 I=∫_𝟎^(𝝅/𝟐)β–’γ€–γ€–πœπ¨π¬γ€—^𝟐⁑ (𝝅/πŸβˆ’π’™)𝒅𝒙〗 I= ∫_𝟎^((𝝅 )/𝟐)▒〖〖𝐬𝐒𝐧〗^𝟐 𝒙〗⁑𝒅𝒙 Using P4 : ∫_0^π‘Žβ–’γ€–π‘“(π‘₯)𝑑π‘₯=γ€— ∫_0^π‘Žβ–’π‘“(π‘Žβˆ’π‘₯)𝑑π‘₯ (As cos (πœ‹/2βˆ’πœƒ)=sinβ‘πœƒ) …(2) Adding (1) and (2) I+I= ∫_0^(πœ‹/2)β–’γ€–cos^2⁑π‘₯ 𝑑π‘₯γ€— + ∫_0^((πœ‹ )/2)β–’γ€–sin^2 π‘₯〗⁑𝑑π‘₯ 2I= ∫_0^((πœ‹ )/2)β–’(cos^2⁑〖π‘₯+sin^2⁑π‘₯ γ€— )⁑𝑑π‘₯ 𝟐𝐈 =∫_𝟎^((𝝅 )/𝟐)β–’γ€–πŸ .〗⁑𝒅𝒙 2I=[π‘₯]_0^(πœ‹/2) 2I =πœ‹/2βˆ’0 2I =πœ‹/2 𝐈=𝝅/πŸ’

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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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.