Check sibling questions

Ex 7.5, 22 - Integrate xdx / (x - 1) (x - 2) equals - Integration by partial fraction - Type 1

Ex 7.5, 22 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.5, 22 - Chapter 7 Class 12 Integrals - Part 3


Transcript

Ex 7.5, 22 ﷮﷮ 𝑥𝑑𝑥﷮(𝑥−1)(𝑥−2)﷯﷯ equals • log 𝑥−1﷯2﷮𝑥−2﷯﷯ + C • log 𝑥−2﷯2﷮𝑥−1﷯﷯ + C • log 𝑥−1﷯2﷮𝑥−2﷯﷯ + C • log 𝑥−1﷯ 𝑥−2﷯﷯ + C We can write integrand as 𝑥﷮ 𝑥 − 1﷯ 𝑥 − 2﷯﷯ = 𝐴﷮ 𝑥 − 1﷯﷯ + 𝐵﷮ 𝑥 − 2﷯﷯ 𝑥﷮ 𝑥 − 1﷯ 𝑥 − 2﷯﷯ = 𝐴 𝑥 − 2﷯ + 𝐵 𝑥 − 1﷯﷮ 𝑥 − 1﷯ 𝑥 − 2﷯﷯ By cancelling denominator 𝑥 = 𝐴 𝑥 − 2﷯ + 𝐵 𝑥 − 1﷯ Putting x = 1 in (1) 1 = 𝐴 1−2﷯+𝐵 1−1﷯ 1 = 𝐴× −1﷯+𝐵×0 1 = −𝐴 𝐴 = − 1 Similarly putting x = 2 in (1) 𝑥 = 𝐴 𝑥 − 2﷯ + 𝐵 𝑥 − 1﷯ 2 = 𝐴 2−2﷯+𝐵 2−1﷯ 2 = 𝐴×0+𝐵×1 2 = 𝐵 𝐵 = 2 Therefore ﷮﷮ 𝑥 𝑑𝑥﷮ 𝑥 − 1﷯ 𝑥 − 2﷯﷯﷯ 𝑑𝑡 = ﷮﷮ −1﷮ 𝑥 − 1﷯ ﷯﷯ 𝑑𝑥 + ﷮﷮ 2﷮ 𝑥 − 2﷯﷯﷯ = − log ﷮ 𝑥−1﷯﷯+2 log ﷮ 𝑥−2﷯﷯+𝐶 = − log ﷮ 𝑥−1﷯﷯+ log ﷮ 𝑥−2﷯﷮2﷯﷯+𝐶 = log ﷮ 𝑥 − 2﷯﷮2﷯﷮𝑥 − 1﷯﷯﷯+𝐶 ∴ The correct answer is B.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.