Ex 7.4, 8 - Integrate x2 / root x6 + a6 - Chapter 7 NCERT

Ex 7.4, 8 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.4, 8 - Chapter 7 Class 12 Integrals - Part 3

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Transcript

Ex 7.4, 8 Integrate 𝑥^2/√(𝑥^6 + 𝑎^6 ) Let 𝑥^3=𝑡 Differentiating both sides w.r.t. x 3𝑥^2=𝑑𝑡/𝑑𝑥 𝑑𝑥=𝑑𝑡/(3𝑥^2 ) Integrating the function ∫1▒𝑥^2/√(𝑥^6 + 𝑎^6 ) 𝑑𝑥=∫1▒𝑥^2/√((𝑥^3 )^2 + (𝑎^3 )^2 ) 𝑑𝑥 Putting values of 𝑥^3=𝑡 and 𝑑𝑥=𝑑𝑡/(3𝑥^2 ) , we get =∫1▒𝑥^2/√(𝑡^2 + (𝑎^3 )^2 ) 𝑑𝑥 =∫1▒𝑥^2/√(𝑡^2 + (𝑎^3 )^2 ) . 𝑑𝑡/(3𝑥^2 ) =∫1▒1/√((𝑡^2 + (𝑎^3 )^2 ) ) . 𝑑𝑡/3 =1/3 ∫1▒𝑑𝑡/√(𝑡^2 + (𝑎^3 )^2 ) =1/3 [log⁡|𝑡+√(𝑡^2 + (𝑎^3 )^2 )|+𝐶1] It is of form ∫1▒𝑑𝑥/√(𝑥^2 + 𝑎^2 ) =log⁡|𝑥+√(𝑥^2 + 𝑎^2 )|+𝐶1 ∴ Replacing 𝑥 by 𝑡 and a by 𝑎^3, we get =1/3 log⁡|𝑡+√(𝑡^2 + 𝑎^6 ) |+𝐶 =1/3 log⁡|𝑥^3+√((𝑥^3 )^2 + 𝑎^6 ) |+𝐶 =𝟏/𝟑 𝒍𝒐𝒈⁡|𝒙^𝟑+√(𝒙^𝟔+ 𝒂^𝟔 ) |+𝑪 ("Using" 𝑡=𝑥^3 )

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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.