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Misc 3 - Chapter 7 Class 12 Integrals

Last updated at May 29, 2023 by

Misc 3 - Integrate 1 / x root ax + x2 - Chapter 7 Class 12

Misc 3 - Chapter 7 Class 12 Integrals - Part 2
Misc 3 - Chapter 7 Class 12 Integrals - Part 3 Misc 3 - Chapter 7 Class 12 Integrals - Part 4

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Misc 3 Integrate the function 1/(𝑥 √(𝑎𝑥 − 𝑥^2 ) ) ∫1▒1/(𝑥 √(𝑎𝑥 − 𝑥^2 ) ) 𝑑𝑥 = ∫1▒1/(𝑥 √(𝑥^2 (𝑎/𝑥 − 1) ) ) 𝑑𝑥 = ∫1▒1/(𝑥 . 𝑥√((𝑎/𝑥 − 1) ) ) 𝑑𝑥 = ∫1▒1/(𝑥^2 √((𝑎/𝑥 − 1) ) ) 𝑑𝑥 Let 𝑎/𝑥 −1=𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 (− 𝑎)/𝑥^2 −0 = 𝑑𝑡/𝑑𝑥 (− 𝑎)/𝑥^2 = 𝑑𝑡/𝑑𝑥 𝑑𝑥 = (− 𝑥^2)/𝑎 . 𝑑𝑡 Putting the values of (a/x−1) and dx, we get ∫1▒1/(𝑥^2 √((𝑎/𝑥 − 1) ) ) 𝑑𝑥 = ∫1▒1/(𝑥^2 √𝑡 ) 𝑑𝑥 = ∫1▒1/(𝑥^2 √𝑡 )×(−𝑥^2)/𝑎 . 𝑑𝑡 = ∫1▒(−1)/𝑎 𝑑𝑡/√𝑡 = (−1)/𝑎 ∫1▒(𝑡)^(− 1/2) 𝑑𝑡 = (−1)/𝑎 [𝑡^((−1)/2 + 1)/((−1)/2 + 1)]+𝐶 = (−1)/𝑎 [𝑡^(1/2)/(1/2)] + 𝐶 = (−2)/𝑎 [√𝑡] + 𝐶 Putting back t = 𝑎/𝑥−1 = (−2)/𝑎 √(𝑎/𝑥 −1) + 𝐶 = (−𝟐)/𝒂 √((𝒂 − 𝒙)/𝒙) + 𝑪

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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.