Ex 7.11, 1 - Chapter 7 Class 12 Integrals (Term 2)

Last updated at Dec. 11, 2021 by Teachoo

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Transcript

Ex 7.11, 1 By using the properties of definite integrals, evaluate the integrals : ∫_0^(𝜋/2)▒〖cos^2⁡𝑥 𝑑𝑥〗 Let 𝐈=∫_𝟎^(𝝅/𝟐)▒〖〖𝒄𝒐𝒔〗^𝟐⁡𝒙 𝒅𝒙〗 I=∫_𝟎^(𝝅/𝟐)▒〖〖𝐜𝐨𝐬〗^𝟐⁡ (𝝅/𝟐−𝒙)𝒅𝒙〗 I= ∫_𝟎^((𝝅 )/𝟐)▒〖〖𝐬𝐢𝐧〗^𝟐 𝒙〗⁡𝒅𝒙 Using P4 : ∫_0^𝑎▒〖𝑓(𝑥)𝑑𝑥=〗 ∫_0^𝑎▒𝑓(𝑎−𝑥)𝑑𝑥 (As cos (𝜋/2−𝜃)=sin⁡𝜃) …(2) Adding (1) and (2) I+I= ∫_0^(𝜋/2)▒〖cos^2⁡𝑥 𝑑𝑥〗 + ∫_0^((𝜋 )/2)▒〖sin^2 𝑥〗⁡𝑑𝑥 2I= ∫_0^((𝜋 )/2)▒(cos^2⁡〖𝑥+sin^2⁡𝑥 〗 )⁡𝑑𝑥 𝟐𝐈 =∫_𝟎^((𝝅 )/𝟐)▒〖𝟏 .〗⁡𝒅𝒙 2I=[𝑥]_0^(𝜋/2) 2I =𝜋/2−0 2I =𝜋/2 𝐈=𝝅/𝟒

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