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Ex 7.5, 12 - Integrate x3 + x+ 1 / x2 - 1 - Chapter 7 CBSE - Integration by partial fraction - Type 1

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.5, 12 𝑥﷮3﷯ + 𝑥 + 1﷮𝑥2− 1﷯ 𝑥﷮3﷯ + 𝑥 + 1﷮𝑥2− 1﷯ = 𝑥+ 2𝑥 + 1﷮ 𝑥﷮2﷯ − 1﷯ = 𝑥+ 2𝑥 + 1﷮ 𝑥 − 1﷯ 𝑥 + 1﷯﷯ Now solving 2𝑥 + 1﷮ 𝑥 − 1﷯ 𝑥 + 1﷯﷯ We can write it as 2𝑥 + 1﷮ 𝑥 − 1﷯ 𝑥 + 1﷯﷯ = 𝐴﷮𝑥 − 1﷯ + 𝐵﷮𝑥 + 1﷯ 2𝑥 + 1﷮ 𝑥 − 1﷯ 𝑥 + 1﷯﷯ = 𝐴 𝑥 + 1﷯ + 𝐵 𝑥 − 1﷯﷮ 𝑥 − 1﷯﷯ By cancelling denominator 2𝑥+1 = 𝐴 𝑥 + 1﷯ + 𝐵 𝑥 − 1﷯ Putting x = 1 , in (1) 2+1 = 𝐴 1+1﷯ + 𝐵 1−1﷯ 3 = 2𝐴+ 𝐵×0 𝐴 = 3﷮2﷯ Similarly Putting x=−1 , in (1) 2 −1﷯+1 = 𝐴 −1+1﷯+𝐵 −1−1﷯ −2+1 = 𝐴×0+𝐵× −2﷯ −1 = −2𝐵 𝐵 = 1﷮2﷯ Hence we can write it as 2𝑥 + 1﷮ 𝑥 − 1﷯ 𝑥 + 1﷯﷯ = 3﷮2﷯﷮ 𝑥 − 1﷯﷯ + 1﷮2﷯﷮𝑥 + 1﷯ Therefore ﷮﷮ 𝑥﷮3﷯ + 𝑥 + 1﷮𝑥2− 1﷯﷯ 𝑑𝑥 = ﷮﷮ 𝑥+ 3﷮2 𝑥 − 1﷯﷯ + 1﷮2 𝑥 + 1﷯﷯ ﷯﷯ 𝑑𝑥 = ﷮﷮𝑥﷯𝑑𝑥+ 3﷮2﷯ ﷮﷮ 𝑑𝑥﷮𝑥 − 1﷯﷯+ 1﷮2﷯ ﷮﷮ 𝑑𝑥﷮𝑥 + 1﷯﷯ 𝑑𝑥 = 𝒙﷮𝟐﷯﷮𝟐﷯ + 𝟑﷮𝟐﷯ 𝐥𝐨𝐠 ﷮ 𝒙−𝟏﷯﷯+ 𝟏﷮𝟐﷯ 𝒍𝒐𝒈 ﷮ 𝒙+𝟏﷯﷯+𝑪

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