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Misc 15 - Integrate cos3 x elog sinx - Chapter 7 NCERT - Miscellaneous

Misc 15 - Chapter 7 Class 12 Integrals - Part 2
Misc 15 - Chapter 7 Class 12 Integrals - Part 3

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Misc 15 Integrate the function cos^3⁑π‘₯ 𝑒^log⁑sin⁑π‘₯ ∫1▒〖𝑒^log⁑sin⁑π‘₯ cos^3〗⁑π‘₯ = ∫1▒〖𝑠𝑖𝑛 π‘₯ cos^3〗⁑〖π‘₯ 𝑑π‘₯γ€— Let t = sin x 𝑑𝑑/𝑑π‘₯=cos⁑π‘₯ 𝑑𝑑/cos⁑π‘₯ = 𝑑π‘₯ (𝑒^logβ‘π‘Ž =π‘Ž) Putting value of t and dt in our equation ∫1▒〖𝑠𝑖𝑛 π‘₯ cos^3〗⁑〖π‘₯ 𝑑π‘₯γ€— = ∫1▒〖𝑑 γ€–π‘π‘œπ‘ γ€—^3 γ€— π‘₯ 𝑑π‘₯ = ∫1▒〖𝑑 γ€–π‘π‘œπ‘ γ€—^3 γ€— π‘₯×𝑑𝑑/cos⁑π‘₯ = ∫1▒〖𝑑 γ€–π‘π‘œπ‘ γ€—^2 γ€— π‘₯ 𝑑𝑑 = ∫1▒𝑑(1βˆ’sin^2⁑π‘₯) 𝑑𝑑 = ∫1▒〖𝑑 (1βˆ’π‘‘^2 γ€—) 𝑑𝑑 = ∫1β–’(π‘‘βˆ’π‘‘^3 ) 𝑑𝑑 = 𝑑^2/2βˆ’π‘‘^4/4+ C Putting back value of 𝑑 = sin x = ((γ€–sin⁑〖π‘₯)γ€—γ€—^2)/2βˆ’(〖𝑠𝑖𝑛〗^4 π‘₯)/4+ C =(1 βˆ’ γ€–π‘π‘œπ‘ γ€—^2 π‘₯)/2βˆ’ (1 βˆ’ γ€–π‘π‘œπ‘ γ€—^2 π‘₯)^2/4+ C = (1 βˆ’γ€– π‘π‘œπ‘ γ€—^2 π‘₯)/4βˆ’((1 + γ€–π‘π‘œπ‘ γ€—^4 βˆ’2γ€–π‘π‘œπ‘ γ€—^2 π‘₯)/4)+ C = 1/2βˆ’(γ€–π‘π‘œπ‘ γ€—^2 π‘₯)/2βˆ’1/4βˆ’(γ€–π‘π‘œπ‘ γ€—^4 π‘₯)/4+(γ€–π‘π‘œπ‘ γ€—^2 π‘₯)/2+ C = ((γ€–sin⁑〖π‘₯)γ€—γ€—^2)/2βˆ’(〖𝑠𝑖𝑛〗^4 π‘₯)/4+ C = (1 βˆ’ γ€–π‘π‘œπ‘ γ€—^2 π‘₯)/2βˆ’ (1 βˆ’ γ€–π‘π‘œπ‘ γ€—^2 π‘₯)^2/4+ C = (1 βˆ’γ€– π‘π‘œπ‘ γ€—^2 π‘₯)/2βˆ’((1 + γ€–π‘π‘œπ‘ γ€—^4 βˆ’ 2γ€–π‘π‘œπ‘ γ€—^2 π‘₯)/4)+ C = 1/2βˆ’(γ€–π‘π‘œπ‘ γ€—^2 π‘₯)/2βˆ’1/4βˆ’(γ€–π‘π‘œπ‘ γ€—^4 π‘₯)/4+(γ€–π‘π‘œπ‘ γ€—^2 π‘₯)/2+ C = 1/4βˆ’(γ€–π‘π‘œπ‘ γ€—^4 π‘₯)/4+ C = (βˆ’γ€–π’„π’π’”γ€—^πŸ’ 𝒙)/πŸ’+ π‘ͺ_𝟏 (Where 𝐢_1=1/4+𝐢) = ((γ€–sin⁑〖π‘₯)γ€—γ€—^2)/2βˆ’(〖𝑠𝑖𝑛〗^4 π‘₯)/4+ C = (1 βˆ’ γ€–π‘π‘œπ‘ γ€—^2 π‘₯)/2βˆ’ (1 βˆ’ γ€–π‘π‘œπ‘ γ€—^2 π‘₯)^2/4+ C = (1 βˆ’γ€– π‘π‘œπ‘ γ€—^2 π‘₯)/2βˆ’((1 + γ€–π‘π‘œπ‘ γ€—^4 βˆ’ 2γ€–π‘π‘œπ‘ γ€—^2 π‘₯)/4)+ C = 1/2βˆ’(γ€–π‘π‘œπ‘ γ€—^2 π‘₯)/2βˆ’1/4βˆ’(γ€–π‘π‘œπ‘ γ€—^4 π‘₯)/4+(γ€–π‘π‘œπ‘ γ€—^2 π‘₯)/2+ C = 1/4βˆ’(γ€–π‘π‘œπ‘ γ€—^4 π‘₯)/4+ C = (βˆ’γ€–π’„π’π’”γ€—^πŸ’ 𝒙)/πŸ’+ π‘ͺ_𝟏

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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.