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Misc 7 - Integrate sin x / sin (x - a) - Chapter 7 Class 12

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

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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 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.