Misc 11 - Integrate 1 / cos (x + a) cos (x + b) - Class 12 - Integration using trigo identities - a-b formulae

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Misc 11 Integrate the function 1﷮ cos﷮ 𝑥 + 𝑎﷯﷯ cos﷮ 𝑥 + 𝑏﷯﷯﷯ ﷮﷮ 𝑑𝑥﷮ cos﷮ 𝑥 + 𝑎﷯ cos﷮(𝑥 + 𝑏)﷯﷯﷯﷯ Divide & Multiplying by 𝐬𝐢𝐧﷮ 𝒂−𝒃﷯﷯ = ﷮﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷯ × 1﷮ cos﷮ 𝑥 + 𝑎﷯﷯ cos﷮ 𝑥 + 𝑏﷯﷯﷯﷯ 𝑑𝑥 = 1﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷯ ﷮﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷮ cos﷮ 𝑥 + 𝑎﷯﷯ cos﷮ 𝑥 + 𝑏﷯﷯﷯﷯ 𝑑𝑥 = 1﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷯ ﷮﷮ sin﷮ 𝑎 − 𝑏 + 𝑥 − 𝑥﷯﷯﷮ cos﷮ 𝑥 + 𝑎﷯﷯ cos﷮ 𝑥 + 𝑏﷯﷯﷯﷯ 𝑑𝑥 = 1﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷯ ﷮﷮ sin ﷮( 𝑥 + 𝑎﷯﷯ − 𝑥 + 𝑏﷯) ﷮ cos﷮ 𝑥 + 𝑎﷯﷯ cos﷮ 𝑥 + 𝑏﷯﷯﷯﷯ 𝑑𝑥 = 1﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷯ ﷮﷮ 𝑠𝑖𝑛﷮ 𝑥 + 𝑎﷯﷯ 𝑐𝑜𝑠﷮ 𝑥 + 𝑏﷯﷯ − 𝑐𝑜𝑠﷮ 𝑥 + 𝑎﷯﷯ 𝑠𝑖𝑛﷮ 𝑥 + 𝑏﷯﷯ ﷮ cos﷮ 𝑥 + 𝑎﷯﷯ cos﷮ 𝑥 + 𝑏﷯﷯﷯﷯ 𝑑𝑥 = 1﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷯ ﷮﷮ 𝑠𝑖𝑛﷮ 𝑥 + 𝑎﷯﷯ 𝑐𝑜𝑠﷮ 𝑥 + 𝑏﷯﷯﷮ cos﷮ 𝑥 + 𝑎﷯﷯ cos﷮ 𝑥 + 𝑏﷯﷯﷯ − cos﷮ 𝑥 + 𝑎﷯﷯ sin﷮ 𝑥 + 𝑏﷯﷯﷮ cos﷮ 𝑥 + 𝑎﷯﷯ cos﷮ 𝑥 + 𝑏﷯﷯﷯﷯﷯ 𝑑𝑥 = 1﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷯ ﷮﷮ 𝑠𝑖𝑛﷮ 𝑥 + 𝑎﷯﷯﷮ cos﷮ 𝑥 + 𝑎﷯﷯﷯ − 𝑠𝑖𝑛﷮ 𝑥 + 𝑏﷯﷯﷮ cos﷮ 𝑥 + 𝑏﷯﷯﷯﷯﷯ 𝑑𝑥 = 1﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷯ ﷮﷮ tan﷮ 𝑥+𝑎﷯﷯ − tan﷮ 𝑥+𝑏﷯﷯﷯﷯ 𝑑𝑥 Thus, our equation becomes ﷮﷮ 1﷮ cos﷮ 𝑥 + 𝑎﷯﷯ cos﷮ 𝑥 + 𝑏﷯﷯﷯﷯ 𝑑𝑥 = 1﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷯ ﷮﷮ tan﷮ 𝑥+𝑎﷯﷯− tan﷮ 𝑥+𝑎﷯﷯﷯ 𝑑𝑥 = 1﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷯ [− log﷮ cos﷮ 𝑥+𝑎﷯﷯﷯﷯+𝐶1− − log﷮ cos﷮ 𝑥+𝑏﷯﷯﷯﷯+𝐶2﷯ = 1﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷯ − log﷮ cos﷮ 𝑥+𝑎﷯﷯﷯﷯+ log﷮ cos﷮ 𝑥+𝑏﷯﷯﷯﷯+𝐶1+𝐶2﷯ = 1﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷯ − log﷮ cos﷮ 𝑥+𝑎﷯﷯﷯﷯+ log﷮ cos﷮ 𝑥+𝑏﷯﷯﷯﷯﷯+ 𝟏﷮ 𝒔𝒊𝒏﷮ 𝒂 + 𝒃﷯﷯﷯ 𝑪𝟏+𝑪𝟐﷯ = 1﷮ sin﷮ 𝑎 − 𝑏﷯﷯﷯ − log﷮ cos﷮ 𝑥+𝑎﷯﷯﷯﷯+ log﷮ cos﷮ 𝑥+𝑏﷯﷯﷯﷯﷯+𝑪 = 𝟏﷮ 𝒔𝒊𝒏﷮ 𝒂 + 𝒃﷯﷯﷯ 𝐥𝐨𝐠 𝒄𝒐𝒔﷮ 𝒙 +𝒃﷯﷯﷮ 𝒄𝒐𝒔﷮ 𝒙 +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 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.