Ex 7.10, 1 - Chapter 7 Class 12 Integrals

Transcript

Ex 7.10, 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 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 Evaluate: ∫_0^𝜋▒〖𝑒^cos⁡𝑥 /(𝑒^cos⁡𝑥 + 𝑒^〖−cos〗⁡𝑥 ) 𝑑𝑥〗 Let I=∫_0^𝜋▒〖𝑒^cos⁡𝑥 /(𝑒^cos⁡𝑥 + 𝑒^〖−cos〗⁡𝑥 ) 𝑑𝑥〗 " " I= ∫_0^𝜋▒〖𝑒^cos⁡〖(𝜋 − 𝑥)〗 /(𝑒^cos⁡〖(𝜋 − 𝑥)〗 + 𝑒^〖−cos〗⁡〖(𝜋 − 𝑥)〗 ) 𝑑𝑥〗 " " I=∫_0^𝜋▒〖𝑒^〖−cos〗⁡𝑥 /(𝑒^〖−cos〗⁡𝑥 + 𝑒^(〖−(−cos〗⁡𝑥)) ) 𝑑𝑥〗 I=∫_0^𝜋▒〖𝑒^〖−cos〗⁡𝑥 /(𝑒^〖−cos〗⁡𝑥 + 𝑒^cos⁡𝑥 ) 𝑑𝑥〗 Evaluate: ∫_0^𝜋▒〖𝑒^cos⁡𝑥 /(𝑒^cos⁡𝑥 + 𝑒^〖−cos〗⁡𝑥 ) 𝑑𝑥〗 Let I=∫_0^𝜋▒〖𝑒^cos⁡𝑥 /(𝑒^cos⁡𝑥 + 𝑒^〖−cos〗⁡𝑥 ) 𝑑𝑥〗 " " I= ∫_0^𝜋▒〖𝑒^cos⁡〖(𝜋 − 𝑥)〗 /(𝑒^cos⁡〖(𝜋 − 𝑥)〗 + 𝑒^〖−cos〗⁡〖(𝜋 − 𝑥)〗 ) 𝑑𝑥〗 " " I=∫_0^𝜋▒〖𝑒^〖−cos〗⁡𝑥 /(𝑒^〖−cos〗⁡𝑥 + 𝑒^(〖−(−cos〗⁡𝑥)) ) 𝑑𝑥〗 I=∫_0^𝜋▒〖𝑒^〖−cos〗⁡𝑥 /(𝑒^〖−cos〗⁡𝑥 + 𝑒^cos⁡𝑥 ) 𝑑𝑥〗 Adding (1) and (2) i.e. (1) + (2) I+I=∫_0^𝜋▒〖𝑒^cos⁡𝑥 /(𝑒^cos⁡𝑥 + 𝑒^〖−cos〗⁡𝑥 ) 𝑑𝑥〗 + ∫_0^𝜋▒〖𝑒^〖−cos〗⁡𝑥 /(𝑒^〖−cos〗⁡𝑥 + 𝑒^cos⁡𝑥 ) 𝑑𝑥〗 2I=∫_0^𝜋▒〖(𝑒^cos⁡𝑥 + 𝑒^〖−cos〗⁡𝑥 )/(𝑒^cos⁡𝑥 + 𝑒^〖−cos〗⁡𝑥 ) 𝑑𝑥〗 2I =∫_0^𝜋▒〖1 .〗⁡𝑑𝑥 2I=[𝑥]_0^𝜋 2I =𝜋−0 2I =𝜋 𝐈=𝝅/𝟐