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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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Ex 7.11, 14 By using the properties of definite integrals, evaluate the integrals : ∫_0^2πœ‹β–’cos^5⁑π‘₯ 𝑑π‘₯ ∫_0^2πœ‹β–’cos^5⁑π‘₯ 𝑑π‘₯ =∫_0^πœ‹β–’cos^5⁑π‘₯ 𝑑π‘₯+∫_0^πœ‹β–’γ€–cos^5 (2Ο€βˆ’π‘₯)γ€— 𝑑π‘₯ = ∫_0^πœ‹β–’γ€–γ€–π‘π‘œπ‘ γ€—^5 π‘₯ 𝑑π‘₯+∫_0^πœ‹β–’γ€–π‘π‘œπ‘ γ€—^5 γ€— π‘₯ = 2 ∫_0^πœ‹β–’γ€–γ€–π‘π‘œπ‘ γ€—^5 π‘₯ 𝑑π‘₯γ€— Using property: ∫_0^2π‘Žβ–’γ€–π‘“(π‘₯)𝑑π‘₯=∫_0^π‘Žβ–’γ€–π‘“(π‘₯)𝑑π‘₯+∫_0^π‘Žβ–’π‘“(2π‘Žβˆ’π‘₯)𝑑π‘₯γ€—γ€— (As cos (2Ο€ βˆ’ πœƒ) = cos πœƒ) Using property: ∫_0^2π‘Žβ–’γ€–π‘“(π‘₯)𝑑π‘₯=∫_0^π‘Žβ–’γ€–π‘“(π‘₯)𝑑π‘₯+∫_0^π‘Žβ–’π‘“(2π‘Žβˆ’π‘₯)𝑑π‘₯γ€—γ€— = 2 (∫_0^(πœ‹/2)β–’γ€–γ€–π‘π‘œπ‘ γ€—^5 π‘₯ 𝑑π‘₯+∫_0^(πœ‹/2)β–’γ€–cos⁑(πœ‹βˆ’π‘₯) γ€—γ€— 𝑑π‘₯) = 2 (∫_0^(πœ‹/2)β–’γ€–γ€–π‘π‘œπ‘ γ€—^5 π‘₯ 𝑑π‘₯+∫_0^(πœ‹/2)β–’γ€–(βˆ’ cos π‘₯)^5⁑𝑑π‘₯ γ€—γ€—) = 2 (∫_0^(πœ‹/2)β–’γ€–γ€–π‘π‘œπ‘ γ€—^5 π‘₯ 𝑑π‘₯βˆ’βˆ«_0^(πœ‹/2)β–’γ€–γ€–γ€–π‘π‘œπ‘ γ€—^5 π‘₯〗⁑𝑑π‘₯ γ€—γ€—) = 2Γ—0 = 0 (cos (πœ‹βˆ’πœƒ) = – cos πœƒ)

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.