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

Transcript

Misc 21 Integrate the function (2 + sin⁑2π‘₯)/(1 + cos⁑2π‘₯ ) 𝑒^π‘₯ We can write integral as ((2 +γ€– sin〗⁑2π‘₯)/(1 + cos⁑2π‘₯ )) 𝑒^π‘₯ = [(2 + sin⁑2π‘₯)/(1 + (2 cos^2⁑〖π‘₯ βˆ’ 1γ€— ) )] 𝑒^π‘₯ = [(2 + sin⁑2π‘₯)/(2 cos^2⁑π‘₯ )] 𝑒^π‘₯ = [(2 + 2 cos⁑〖π‘₯ sin⁑π‘₯ γ€—)/(2 cos^2⁑π‘₯ )]𝑒π‘₯ = [2(1 +γ€– cos〗⁑〖π‘₯ sin⁑π‘₯ γ€— )/(2 cos^2⁑π‘₯ )]𝑒π‘₯ (cos⁑2π‘₯=2 cos^2⁑〖π‘₯βˆ’1γ€— ) (sin⁑2π‘₯=2 cos⁑〖π‘₯ sin⁑π‘₯ γ€—) = [(1 + cos⁑〖π‘₯ sin⁑π‘₯ γ€—)/cos^2⁑π‘₯ ]𝑒π‘₯ = [1/cos^2⁑π‘₯ +cos⁑〖π‘₯ sin⁑π‘₯ γ€—/cos^2⁑π‘₯ ] 𝑒^π‘₯ = [sec^2⁑〖π‘₯+cos⁑〖π‘₯ sin⁑π‘₯ γ€—/cos⁑〖π‘₯ cos⁑π‘₯ γ€— γ€— ] 𝑒^π‘₯ = [sec^2⁑〖π‘₯+tan⁑π‘₯ γ€— ] 𝑒^π‘₯ = [tan⁑〖π‘₯+sec^2⁑π‘₯ γ€— ] 𝑒^π‘₯ It is of the form ∫1▒〖𝑒^π‘₯ [𝑓(π‘₯)+𝑓^β€² (π‘₯)] γ€— 𝑑π‘₯=𝑒^π‘₯ 𝑓(π‘₯)+𝐢 Where 𝑓(π‘₯)=tan⁑π‘₯ 𝑓^β€² (π‘₯)=sec^2⁑π‘₯ So, our equation becomes ∫1β–’γ€–[(2 + sin⁑2π‘₯)/(1 + cos⁑2π‘₯ )] 𝑒^π‘₯ 𝑑π‘₯=∫1▒〖𝑒^π‘₯ [tan⁑〖π‘₯+sec^2⁑π‘₯ γ€— ]𝑑π‘₯γ€—γ€— =𝒆^𝒙 𝒕𝒂𝒏⁑𝒙+𝐂

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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.