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Misc 16 - Integrate e3 log x (x4 + 1)-1 - Class 12 CBSE - Miscellaneous

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Misc 16 Integrate the function 𝑒﷮3 log﷮𝑥﷯﷯ 𝑥﷮4﷯+1﷯﷮−1﷯ Hence we can write 𝑒﷮3 log﷮𝑥﷯﷯ 𝑥﷮4﷯+1﷯﷮−1﷯= 𝑒﷮ log﷮𝑥﷯﷮3﷯﷯ 𝑥﷮4﷯+1﷯﷮−1﷯ = 𝑥﷮3﷯ 𝑥﷮4﷯+1﷯﷮−1﷯ Integrating 𝑤.𝑟.𝑡.𝑥 ﷮﷮ 𝑒﷮3 log﷮𝑥﷯﷯ 𝑥﷮4﷯+1﷯﷮−1﷯﷯ = ﷮﷮ 𝑥﷮3﷯ 𝑥﷮4﷯+1﷯﷮−1﷯ 𝑑𝑥﷯ Put t = x﷮4﷯+1﷯ Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑𝑡﷮𝑑𝑥﷯ = 4 𝑥﷮3﷯ 𝑑𝑡﷮4 𝑥﷮3﷯﷯ = 𝑑𝑥 Therefore we can write ﷮﷮ 𝑥﷮3﷯ 𝑥﷮4﷯+1﷯﷮−1﷯ 𝑑𝑥﷯= ﷮﷮ 𝑥﷮3﷯ 𝑡﷯﷮−1﷯ 𝑑𝑡﷮4 𝑥﷮3﷯﷯﷯ = 1﷮4﷯ ﷮﷮ 𝑡﷮−1﷯ 𝑑𝑡﷯ = 1﷮4﷯ ﷮﷮ 𝑑𝑡﷮𝑡﷯﷯ = 1﷮4﷯ log ﷮ 𝑡﷯﷯+𝐶 Putting t = 𝑥﷮4﷯+1 = 1﷮4﷯ log ﷮ 𝑥﷮4﷯+1﷯﷯+C = 𝟏﷮𝟒﷯ log ( 𝒙﷮𝟒﷯+𝟏) + 𝐂

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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