Check sibling questions

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

Maths Crash Course - Live lectures + all videos + Real time Doubt solving!


Transcript

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)

Ask a doubt (live)
Davneet Singh's photo - Co-founder, Teachoo

Made by

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.