Integration Full Chapter Explained - Integration Class 12 - Everything you need

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

Transcript

Misc 7 Integrate the function sin⁑π‘₯/sin⁑(π‘₯ βˆ’ π‘Ž) Let I = ∫1β–’sin⁑π‘₯/sin⁑(π‘₯ βˆ’ π‘Ž) 𝑑π‘₯ Put t = π‘₯ βˆ’ π‘Ž Differentiating 𝑀.π‘Ÿ.𝑑.π‘₯ 𝑑𝑑/𝑑π‘₯ = 𝑑(π‘₯ βˆ’ π‘Ž)/𝑑π‘₯ 𝑑𝑑/𝑑π‘₯ = 1 𝑑π‘₯ = 𝑑𝑑 Therefore ∫1β–’γ€–sin 〗⁑(𝑑 + π‘Ž)/sin⁑𝑑 𝑑𝑑 = ∫1β–’(sin⁑𝑑 cosβ‘π‘Ž + cos⁑𝑑 sinβ‘π‘Ž)/sin⁑𝑑 𝑑𝑑 = ∫1β–’((sin⁑𝑑 cosβ‘π‘Ž)/sin⁑𝑑 + (cos⁑𝑑 sinβ‘π‘Ž)/sin⁑𝑑 ) 𝑑𝑑 = ∫1β–’cosβ‘π‘Ž 𝑑𝑑 + ∫1β–’π‘π‘œπ‘‘β‘π‘‘ sinβ‘π‘Ž 𝑑𝑑 = cosβ‘π‘Ž ∫1▒𝑑𝑑 + sinβ‘π‘Ž ∫1β–’cot⁑𝑑 𝑑𝑑 = cos a Γ— t + sin a log |sin⁑𝑑 | + C Putting back t = x – a = (x – a) cos a + sin a |sin⁑〖(π‘₯βˆ’π‘Ž)γ€— | + C sin⁑(𝐴+𝐡) =sin⁑𝐴 cos⁑𝐡+cos⁑𝐴 sin⁑𝐡 (𝑆𝑖𝑛𝑐𝑒 π‘ π‘–π‘›β‘π‘Ž,π‘π‘œπ‘ β‘π‘Ž π‘Žπ‘Ÿπ‘’ π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘π‘ ) = sin a log |sin⁑〖(π‘₯βˆ’π‘Ž)γ€— | + x cos a βˆ’ a cos a + C = sin a log |𝐬𝐒𝐧⁑〖(π’™βˆ’π’‚)γ€— | + x cos a + 𝐂_𝟏 (𝐢_1= βˆ’ a cos a + 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.