Misc 17 - Integrate f' (ax + b) [f(ax + b)]n - Class 12 - Integration by substitution - x^n

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Misc 17 Integrate the function 𝑓﷮′﷯ 𝑎𝑥+𝑏﷯ 𝑓 𝑎𝑥+𝑏﷯﷯﷮𝑛﷯ Let 𝐼 = ﷮﷮ 𝑓﷮′﷯ 𝑎𝑥+𝑏﷯ 𝑓 𝑎𝑥+𝑏﷯﷯﷮𝑛﷯𝑑𝑥﷯ Put 𝑡=𝑓 𝑎𝑥+𝑏﷯ Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑𝑡﷮𝑑𝑥﷯ = 𝑑 𝑓 𝑎𝑥 + 𝑏﷯﷯﷮𝑑𝑥﷯ 𝑑𝑡﷮𝑑𝑥﷯ = 𝑓﷮′﷯ 𝑎𝑥 + 𝑏﷯× 𝑑﷮𝑑𝑥﷯ 𝑎𝑥 + 𝑏﷯ 𝑑𝑡﷮𝑑𝑥﷯ = 𝑓﷮′﷯ 𝑎𝑥 + 𝑏﷯×𝑎 𝑑𝑡﷮𝑑𝑥﷯ = a 𝑓﷮′﷯ 𝑎𝑥+𝑏﷯ 𝑑𝑡﷮a 𝑓﷮′﷯ 𝑎𝑥 + 𝑏﷯﷯ = 𝑑𝑥 Therefore we can write 𝐼 = ﷮﷮ 𝑓 𝑎𝑥+𝑏﷯﷯﷮𝑛﷯ 𝑓﷮′﷯ 𝑎𝑥+𝑏﷯﷯ 𝑑𝑥 = ﷮﷮ 𝑡﷯﷮𝑛﷯ 𝑓﷮′﷯ 𝑎𝑥+𝑏﷯﷯ 𝑑𝑡﷮a 𝑓﷮′﷯ 𝑎𝑥 + 𝑏﷯﷯ = ﷮﷮ 𝑡﷮𝑛﷯﷮a﷯﷯ 𝑑𝑡 = 1﷮𝑎﷯ ﷮﷮ 𝑡﷮𝑛﷯﷯ 𝑑𝑡 = 1﷮𝑎﷯ 𝑡﷮𝑛 + 1﷯﷮𝑛 + 1﷯ +𝐶 Putting back t =𝑓 𝑎𝑥+𝑏﷯ = 1﷮𝑎﷯ 𝑓 𝑎𝑥 + 𝑏﷯﷯﷮𝑛 + 1﷯﷮𝑛 + 1﷯ +𝐶 = 𝒇 𝒂𝒙+𝒃﷯﷯﷮𝒏 + 𝟏﷯﷮𝒂 𝒏 + 𝟏﷯﷯ +𝑪

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