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Misc 10 - Integrate sin8 x - cos8 x / 1 - 2 sin2 x cos2 x

Misc 10 - Chapter 7 Class 12 Integrals - Part 2
Misc 10 - Chapter 7 Class 12 Integrals - Part 3

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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 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.