Chapter 7 Class 12 Integrals
Concept wise

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Transcript

Ex 7.9, 9 The value of the integral ∫_(1/3)^1▒〖 (𝑥 −𝑥^3 )^(1/3)/𝑥^4 〗 𝑑𝑥 is 6 (B) 0 (C) 3 (D) 4 ∫_(1/3)^1▒〖 (𝑥 − 𝑥^3 )^(1/3)/𝑥^4 〗 𝑑𝑥 Taking common 𝑥^3 from numerator = ∫_(1/3)^1▒〖 ((𝑥^3 )^(1/3) (1/𝑥^2 −1)^(1/3))/𝑥^4 〗 𝑑𝑥 = ∫_(1/3)^1▒〖 (𝑥 (1/𝑥^2 −1)^(1/3))/𝑥^4 〗 𝑑𝑥 = ∫_(1/3)^1▒〖 ( (1/𝑥^2 −1)^(1/3))/𝑥^3 〗 𝑑𝑥 Let t = 1/𝑥^2 −1 𝑑𝑡/𝑑𝑥=(−2)/𝑥^3 (−𝑑𝑡)/2=𝑑𝑥/𝑥^3 Thus, when x varies from 1/3 to 1, t varies form 0 to 8 Substituting values, ∫_(1/3)^1▒〖 ( (1/𝑥^2 −1)^(1/3))/𝑥^3 〗 𝑑𝑥 = 1/2 ∫_8^0▒〖𝑡^(1/3) 𝑑𝑡〗 = (−1)/2 [𝑡^(1/3 + 1)/(1/3 + 1)]_8^0 = (−1)/2 [〖3𝑡〗^(4/3 )/4]_8^0 Putting limits = (−1)/2 (0−(3(8)^(4/3))/4) = 1/2 (3/4) (8)^(4/3) = 1/2 (3/4) (2^3 )^(4/3) = 1/2 (3/4) (2^4 ) = 6 So, (A) is the correct answer.

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.