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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


Transcript

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

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.