Ex 7.5, 3 - Integrate 3x - 1 / (x - 1) (x - 2) (x - 3) - Integration by partial fraction - Type 3

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.5, 3 3𝑥 − 1﷮ 𝑥 − 1﷯ 𝑥 − 2﷯ 𝑥 − 3﷯﷯ We can write the integrand as 3𝑥 − 1﷮ 𝑥 − 1﷯ 𝑥 − 2﷯ 𝑥 − 3﷯﷯ = 𝐴﷮ 𝑥 − 1﷯﷯ + 𝐵﷮ 𝑥 − 2﷯﷯ + 𝐶﷮ 𝑥 − 3﷯﷯ = 𝐴 𝑥 − 2﷯ 𝑥 − 3﷯ + 𝐵 𝑥 − 1﷯ 𝑥 − 3﷯ + 𝐶 𝑥 −1﷯ 𝑥 − 2﷯﷮ 𝑥 − 1﷯ 𝑥 − 2﷯ 𝑥 − 3﷯﷯ By cancelling denominator 3𝑥−1 = 𝐴 𝑥−2﷯ 𝑥−3﷯+𝐵 𝑥−1﷯ 𝑥−3﷯+𝐶 𝑥−1﷯ 𝑥−2﷯ Putting 𝑥=1 in (1) 3−1 = 𝐴 1−2﷯ 1−3﷯+𝐵 1−1﷯ 1−3﷯+𝐶 1−1﷯ 1−2﷯ 2 = 𝐴 −1﷯ −2﷯+𝐵×0+𝐶×0 2 = 2𝐴 𝐴 = 1 Similarly putting 𝑥=2 , in (1) 3 2﷯−1 = 𝐴 2−2﷯ 2−3﷯+𝐵 2−1﷯ 2−3﷯+𝐶 2−1﷯ 2−2﷯ 6−1 = 𝐴×0+𝐵 1﷯ −1﷯+𝐶×0 5 = −𝐵 𝐵 = −5 Similarly putting x = 3 , in (1) 3×3−1 = 𝐴 3−2﷯ 3−3﷯+𝐵 3−1﷯ 3−3﷯+𝐶 3−1﷯ 3−2﷯ 9−1 = 𝐴×0+𝐵×0+𝐶× 2﷯ 1﷯ 8 = 2𝐶 𝐶 = 4 Hence we can write it as ﷮﷮ 3𝑥 − 1﷮ 𝑥 − 1﷯ 𝑥 − 2﷯ 𝑥 − 3﷯﷯﷯ = ﷮﷮ 1﷮𝑥 − 1﷯ + −5﷮𝑥 − 2﷯ + 4﷮𝑥 − 3﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮𝑥 − 1﷯﷯ 𝑑𝑥−5 ﷮﷮ 1﷮𝑥 − 2﷯﷯ 𝑑𝑥+4 ﷮﷮ 1﷮𝑥 − 3﷯﷯ 𝑑𝑥 = 𝐥𝐨𝐠 ﷮ 𝒙−𝟏﷯﷯−𝟓 𝐥𝐨𝐠 ﷮ 𝒙−𝟐﷯﷯+𝟒 𝐥𝐨𝐠 ﷮ 𝒙−𝟑﷯﷯+𝑪

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