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Chapter 7 Class 12 Integrals
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Ex 7.11, 14 - Using properties, evaluate integral cos5 x dx

Ex 7.11, 14 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.11, 14 - Chapter 7 Class 12 Integrals - Part 3

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Transcript

Ex 7.11, 14 By using the properties of definite integrals, evaluate the integrals : ∫_0^2𝜋▒cos^5⁡𝑥 𝑑𝑥 ∫_0^2𝜋▒cos^5⁡𝑥 𝑑𝑥 =∫_0^𝜋▒cos^5⁡𝑥 𝑑𝑥+∫_0^𝜋▒〖cos^5 (2π−𝑥)〗 𝑑𝑥 = ∫_0^𝜋▒〖〖𝑐𝑜𝑠〗^5 𝑥 𝑑𝑥+∫_0^𝜋▒〖𝑐𝑜𝑠〗^5 〗 𝑥 = 2 ∫_0^𝜋▒〖〖𝑐𝑜𝑠〗^5 𝑥 𝑑𝑥〗 Using property: ∫_0^2𝑎▒〖𝑓(𝑥)𝑑𝑥=∫_0^𝑎▒〖𝑓(𝑥)𝑑𝑥+∫_0^𝑎▒𝑓(2𝑎−𝑥)𝑑𝑥〗〗 (As cos (2π − 𝜃) = cos 𝜃) Using property: ∫_0^2𝑎▒〖𝑓(𝑥)𝑑𝑥=∫_0^𝑎▒〖𝑓(𝑥)𝑑𝑥+∫_0^𝑎▒𝑓(2𝑎−𝑥)𝑑𝑥〗〗 = 2 (∫_0^(𝜋/2)▒〖〖𝑐𝑜𝑠〗^5 𝑥 𝑑𝑥+∫_0^(𝜋/2)▒〖cos⁡(𝜋−𝑥) 〗〗 𝑑𝑥) = 2 (∫_0^(𝜋/2)▒〖〖𝑐𝑜𝑠〗^5 𝑥 𝑑𝑥+∫_0^(𝜋/2)▒〖(− cos 𝑥)^5⁡𝑑𝑥 〗〗) = 2 (∫_0^(𝜋/2)▒〖〖𝑐𝑜𝑠〗^5 𝑥 𝑑𝑥−∫_0^(𝜋/2)▒〖〖〖𝑐𝑜𝑠〗^5 𝑥〗⁡𝑑𝑥 〗〗) = 2×0 = 0 (cos (𝜋−𝜃) = – cos 𝜃)

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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.