Find : ∫√(x/(1-x^3 )) dx; x∈(0,1)

This question is similar to Misc 12 Chapter 7 Class 12




โˆซโˆš(๐‘ฅ/(1 โˆ’ ๐‘ฅ^3 )) ๐‘‘๐‘ฅ = โˆซ1โ–’โˆš๐‘ฅ/โˆš(1 โˆ’ ๐‘ฅ^3 ) ๐‘‘๐‘ฅ Let ๐’•=๐’™^(๐Ÿ‘/๐Ÿ) Differentiating w.r.t. ๐‘ฅ ๐‘‘๐‘ก/๐‘‘๐‘ฅ=3/2 ๐‘ฅ^(3/2 โˆ’1) ๐‘‘๐‘ก/๐‘‘๐‘ฅ=3/2 ๐‘ฅ^(1/2 ) ๐‘‘๐‘ก/๐‘‘๐‘ฅ=3/2 โˆš๐‘ฅ ๐Ÿ/๐Ÿ‘ ๐๐ญ=โˆš๐’™ ๐’…๐’™ Now, our equation becomes โˆซ1โ–’โˆš๐‘ฅ/โˆš(1 โˆ’ ๐‘ฅ^3 ) ๐‘‘๐‘ฅ=โˆซ1โ–’ใ€–2/(3โˆš(1 โˆ’ใ€– ๐‘ฅใ€—^3 )) ๐‘‘๐‘กใ€— = ๐Ÿ/๐Ÿ‘ โˆซ1โ–’๐’…๐’•/โˆš(๐Ÿ โˆ’ ๐’•^๐Ÿ )dt = 2/3 sin^(โˆ’1)โกใ€–๐‘ก+๐‘ใ€— Putting back ๐‘ก=๐‘ฅ^(3/2) = ๐Ÿ/๐Ÿ‘ ใ€–๐’”๐’Š๐’ใ€—^(โˆ’๐Ÿ)โกใ€–ใ€–(๐’™ใ€—^(๐Ÿ‘/๐Ÿ))+๐’„ใ€—

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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 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.