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Ex 7.5

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Ex 7.5, 22 (MCQ) You are here

Ex 7.5, 23 (MCQ) Important

Last updated at May 29, 2023 by Teachoo

Ex 7.5, 22 𝑥𝑑𝑥(𝑥−1)(𝑥−2) equals • log 𝑥−12𝑥−2 + C • log 𝑥−22𝑥−1 + C • log 𝑥−12𝑥−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−22+𝐶 = log 𝑥 − 22𝑥 − 1+𝐶 ∴ The correct answer is B.