Misc 22 - Integrate x2 + x + 1 / (x + 1)2 (x + 2) - Miscellaneous

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Misc 22 Integrate the function 𝑥﷮2﷯ + 𝑥 + 1﷮ 𝑥 + 1﷯﷮2﷯ 𝑥 + 2﷯﷯ ﷮﷮ 𝑥﷮2﷯ + 𝑥 + 1﷮ 𝑥 + 1﷯﷮2﷯ 𝑥 + 2﷯﷯ 𝑑𝑥﷯ By partial fraction 𝑥﷮2﷯ + 𝑥 + 1﷮ 𝑥 + 1﷯﷮2﷯ 𝑥 + 2﷯﷯= A﷮𝑥 + 2﷯+ B﷮𝑥 + 1﷯+ C﷮ (𝑥 + 1)﷮2﷯﷯ 𝑥﷮2﷯ + 𝑥 + 1﷮ 𝑥 + 1﷯﷮2﷯ 𝑥 + 2﷯﷯= A (𝑥 + 1)﷮2﷯ + B 𝑥 + 1﷯(𝑥 + 2)+ C(𝑥 + 2)﷮ (𝑥 + 1)﷮2﷯ 𝑥 + 2﷯﷯ Cancelling denominators 𝑥﷮2﷯+𝑥+1=A (𝑥+1)﷮2﷯+B 𝑥+2﷯ 𝑥+1﷯+C(𝑥+2) Hence, 𝑥﷮2﷯ + 𝑥 + 1﷮ 𝑥 + 1﷯﷮2﷯(𝑥 + 2)﷯= 3﷮𝑥 +2﷯− 2﷮𝑥 +1﷯+ 1﷮ (𝑥 + 1)﷮2﷯﷯ ﷮﷮ 𝑥﷮2﷯+ 𝑥 +1﷮ (𝑥 + 1)﷮2﷯ (𝑥 + 2)﷯﷯= ﷮﷮ 3 𝑑𝑥﷮𝑥 + 2﷯﷯− ﷮﷮ 2 𝑑𝑥﷮𝑥 + 1﷯﷯+ ﷮﷮ 1 𝑑𝑥﷮ 𝑥 + 1﷯﷮2﷯﷯﷯ = 3log 𝑥+2﷯ – 2log 𝑥+1﷯− 1﷮𝑥 + 1﷯+𝐶 = – 2log 𝑥+1﷯− 1﷮𝑥 + 1﷯+3log 𝑥+2﷯+𝐶

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