Integration Full Chapter Explained -



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


Ex 7.3, 5 Integrate sin^3⁑π‘₯ cos^3 π‘₯ ∫1β–’γ€–sin^3⁑π‘₯ cos^3 π‘₯γ€— 𝑑π‘₯ =∫1▒〖𝑠𝑖𝑛⁑π‘₯. sin^2⁑π‘₯ cos^3 π‘₯γ€— 𝑑π‘₯ =∫1▒〖𝑠𝑖𝑛⁑π‘₯ (1βˆ’cos^2⁑π‘₯ ) cos^3 π‘₯γ€— 𝑑π‘₯ =∫1β–’γ€–(1βˆ’cos^2⁑π‘₯ ) cos^3 π‘₯γ€—. 𝑠𝑖𝑛⁑π‘₯ 𝑑π‘₯ Let cos⁑π‘₯=𝑑 Differentiating w.r.t.x βˆ’sin⁑π‘₯=𝑑𝑑/𝑑π‘₯ 𝑑π‘₯=𝑑𝑑/(βˆ’sin⁑π‘₯ ) (〖𝑠𝑖𝑛〗^2β‘πœƒ=1βˆ’γ€–π‘π‘œπ‘ γ€—^2β‘πœƒ) …(1) Thus, our equation becomes ∫1β–’γ€–sin^2⁑π‘₯ cos^3 π‘₯γ€— 𝑑π‘₯ =∫1β–’γ€–(1βˆ’cos^2⁑π‘₯ ) cos^3 π‘₯γ€—. 𝑠𝑖𝑛⁑π‘₯ 𝑑π‘₯" " =∫1β–’γ€–(1βˆ’π‘‘^2 ) 𝑑^3 γ€—. 𝑠𝑖𝑛⁑π‘₯Γ—1/(βˆ’sin⁑π‘₯ ) 𝑑𝑑" " =∫1β–’γ€–βˆ’(1βˆ’π‘‘^2 ) 𝑑^3 γ€— 𝑑𝑑" " =βˆ’βˆ«1β–’γ€–(𝑑^3βˆ’π‘‘^5 ) γ€— 𝑑𝑑" " =βˆ’[∫1▒𝑑^3 π‘‘π‘‘βˆ’βˆ«1▒𝑑^5 𝑑𝑑] =βˆ’[𝑑^(3 + 1)/(3 + 1) βˆ’ 𝑑^(5 + 1)/(5 + 1)]+𝐢 =βˆ’[𝑑^4/4 βˆ’ 𝑑^6/6]+𝐢 =γ€–βˆ’π‘‘γ€—^4/4 + 𝑑^6/6+𝐢 =𝑑^6/6 βˆ’ 𝑑^4/4 +𝐢 Putting back value of 𝑑=π‘π‘œπ‘ β‘π‘₯ =(〖𝒄𝒐𝒔〗^πŸ” 𝒙)/πŸ” βˆ’ (〖𝒄𝒐𝒔〗^πŸ’ 𝒙)/πŸ’ +π‘ͺ

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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.