Integration Full Chapter Explained -



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


Ex 7.3, 14 Integrate the function cos⁑〖π‘₯ βˆ’ sin⁑π‘₯ γ€—/(1 + sin⁑2π‘₯ ) ∫1β–’cos⁑〖π‘₯ βˆ’ sin⁑π‘₯ γ€—/(1 + sin⁑2π‘₯ ) 𝑑π‘₯ =∫1β–’cos⁑〖π‘₯ βˆ’γ€– sin〗⁑π‘₯ γ€—/(𝟏 + 2 sin⁑π‘₯ cos⁑π‘₯ ) 𝑑π‘₯ =∫1β–’cos⁑〖π‘₯ βˆ’γ€– sin〗⁑π‘₯ γ€—/(〖𝐬𝐒𝐧〗^πŸβ‘π’™ + γ€–πœπ¨π¬γ€—^πŸβ‘π’™ + 2 sin⁑cos⁑π‘₯ ) 𝑑π‘₯ =∫1β–’cos⁑〖π‘₯ βˆ’γ€– sin〗⁑π‘₯ γ€—/(sin⁑π‘₯ + cos⁑π‘₯ )^2 𝑑π‘₯ Let sin⁑π‘₯+cos⁑π‘₯=𝑑 Differentiating w.r.t.x (𝑠𝑖𝑛⁑2 πœƒ=2 π‘ π‘–π‘›β‘πœƒ π‘π‘œπ‘ β‘πœƒ) (As 〖𝑠𝑖𝑛〗^2β‘πœƒ+γ€–π‘π‘œπ‘ γ€—^2β‘πœƒ=1) 𝑑(sin⁑π‘₯ + cos⁑π‘₯ )/𝑑π‘₯=𝑑𝑑/𝑑π‘₯ cos π‘₯βˆ’sin⁑π‘₯=𝑑𝑑/𝑑π‘₯ 𝑑π‘₯=𝑑𝑑/(cos π‘₯ βˆ’ sin⁑π‘₯ ) Thus, our equation becomes ∫1β–’((cos⁑〖π‘₯ βˆ’ sin⁑π‘₯ γ€— ))/(sin⁑π‘₯ + cos⁑π‘₯ )^2 𝑑π‘₯ =∫1β–’((cos⁑〖π‘₯ βˆ’ sin⁑π‘₯ γ€— ))/𝑑^2 ×𝑑𝑑/cos⁑〖π‘₯ βˆ’ sin⁑π‘₯ γ€— =∫1▒𝑑𝑑/𝑑^2 =∫1▒𝑑^(βˆ’2) 𝑑𝑑 =𝑑^(βˆ’2 +1)/(βˆ’2 + 1) +𝐢 =𝑑^(βˆ’1)/(βˆ’1) +𝐢 =(βˆ’1)/𝑑 +𝐢 Putting value of 𝑑=𝑠𝑖𝑛⁑π‘₯+π‘π‘œπ‘  π‘₯ =(βˆ’πŸ)/(𝐬𝐒𝐧⁑𝒙 + 𝒄𝒐𝒔⁑𝒙 ) +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.