Integration Full Chapter Explained - https://you.tube/Integration-Class-12

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise

Transcript

Misc 10 Integrate the function (γ€–sin^8 π‘₯γ€—β‘βˆ’ cos^8⁑π‘₯)/(1 βˆ’ 2 sin^2⁑〖π‘₯ cos^2⁑π‘₯ γ€— ) ∫1β–’(γ€–sin^8 π‘₯γ€—β‘βˆ’ cos^8⁑π‘₯)/(1 βˆ’ 2 sin^2⁑〖π‘₯ cos^2⁑π‘₯ γ€— ) =∫1β–’((sin^4 π‘₯)^2β‘γ€–βˆ’ γ€— (cos^4 π‘₯)^2)/(1 βˆ’ 2 sin^2⁑〖π‘₯ cos^2⁑π‘₯ γ€— ) =∫1β–’((sin^4 π‘₯ + cos^4⁑π‘₯ )⁑(sin^4⁑π‘₯ βˆ’ cos^4⁑π‘₯ )𝑑π‘₯)/(1 βˆ’ 2 sin^2⁑〖π‘₯ cos^2⁑π‘₯ γ€— ) =∫1β–’((sin^4 π‘₯ + cos^4⁑π‘₯ )⁑〖 ((sin^2 π‘₯)^2 βˆ’ (cos^2 π‘₯)^2 )γ€— 𝑑π‘₯)/(1 βˆ’ 2 sin^2⁑〖π‘₯ cos^2⁑π‘₯ γ€— ) =∫1β–’((sin^4 π‘₯ + cos^4⁑π‘₯ )⁑(sin^2⁑π‘₯ + cos^2⁑π‘₯ ) (sin^2⁑π‘₯ βˆ’ cos^2⁑π‘₯ )𝑑π‘₯)/(1 βˆ’ 2 sin^2⁑〖π‘₯ cos^2⁑π‘₯ γ€— ) =∫1β–’((sin^4 π‘₯ + cos^4⁑π‘₯ )⁑(sin^2⁑π‘₯ + cos^2⁑π‘₯ ) (sin^2⁑π‘₯ βˆ’ cos^2⁑π‘₯ )𝑑π‘₯)/(1 βˆ’ 2 sin^2⁑〖π‘₯ cos^2⁑π‘₯ γ€— ) =∫1β–’(γ€–(sin^4 π‘₯ + cos^4⁑π‘₯ ) (1)〗⁑〖 (sin^2⁑π‘₯ βˆ’ cos^2⁑π‘₯ )γ€— 𝑑π‘₯)/(1 βˆ’ 2 sin^2⁑〖π‘₯ cos^2⁑π‘₯ γ€— ) Adding & Subtracting 2 sin^2⁑π‘₯ cos^2⁑π‘₯ =∫1β–’((sin^4 π‘₯ + cos^4⁑π‘₯ + 2 sin^2⁑π‘₯ cos^2⁑π‘₯ βˆ’ 2 sin^2⁑cos^2⁑π‘₯ )⁑〖 (sin^2⁑π‘₯ βˆ’ cos^2⁑π‘₯ )γ€— 𝑑π‘₯)/(1 βˆ’ 2 sin^2⁑〖π‘₯ cos^2⁑π‘₯ γ€— ) =∫1β–’((((sin^2⁑π‘₯ )^2+ (cos^2⁑π‘₯ )^2 + 2 sin^2⁑π‘₯ cos^2⁑π‘₯ )βˆ’2 sin^2⁑π‘₯ cos^2⁑π‘₯ )⁑(sin^2⁑π‘₯ βˆ’ cos^2⁑π‘₯ )𝑑π‘₯)/(1 βˆ’ 2 sin^2⁑〖π‘₯ cos^2⁑π‘₯ γ€— ) =∫1β–’(γ€–((sin^2⁑π‘₯ + cos^2⁑π‘₯ )^2 βˆ’ 2 sin^2⁑π‘₯ cos^2⁑π‘₯ ) 〗⁑(sin^2⁑π‘₯ βˆ’ cos^2⁑π‘₯ )𝑑π‘₯)/(1 βˆ’ 2 sin^2⁑〖π‘₯ cos^2⁑π‘₯ γ€— ) =∫1β–’(γ€–(1^2 βˆ’ 2 sin^2⁑π‘₯ cos^2⁑π‘₯ ) 〗⁑(sin^2⁑π‘₯ βˆ’ cos^2⁑π‘₯ )𝑑π‘₯)/(1 βˆ’ 2 sin^2⁑〖π‘₯ cos^2⁑π‘₯ γ€— ) =∫1β–’(γ€–(1 βˆ’ 2 sin^2⁑π‘₯ cos^2⁑π‘₯ ) 〗⁑(sin^2⁑π‘₯ βˆ’ cos^2⁑π‘₯ )𝑑π‘₯)/(1 βˆ’ 2 sin^2⁑〖π‘₯ cos^2⁑π‘₯ γ€— ) =∫1β–’(sin^2⁑π‘₯βˆ’cos^2⁑π‘₯ ) 𝑑π‘₯ =βˆ’βˆ«1β–’(cos^2⁑π‘₯βˆ’sin^2⁑π‘₯ ) 𝑑π‘₯ =βˆ’βˆ«1β–’cos⁑2π‘₯ . 𝑑π‘₯ =(βˆ’πŸ)/𝟐 π¬π’π§β‘πŸπ’™+π‘ͺ (sin^2⁑π‘₯ + cos^2⁑π‘₯=1" " ) (Using cos 2πœƒ=γ€–π‘π‘œπ‘ γ€—^2 πœƒβˆ’γ€–π‘ π‘–π‘›γ€—^2 πœƒ)

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.