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Ex 7.2

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Last updated at May 29, 2023 by Teachoo

Ex 7.2, 34 Integrate √(tan𝑥 )/sin〖𝑥 cos𝑥 〗 Simplifying the function √(tan𝑥 )/sin〖𝑥 cos𝑥 〗 = √(tan𝑥 )/(sin〖𝑥 cos𝑥 〗. cos𝑥/cos𝑥 ) = √(tan𝑥 )/(sin𝑥 . cos^2𝑥/cos𝑥 ) = √(tan𝑥 )/(cos^2𝑥 . (sin 𝑥)/cos𝑥 ) Concept: There are two methods to deal with 𝑡𝑎𝑛𝑥 (1) Convert into 𝑠𝑖𝑛𝑥 and 𝑐𝑜𝑠𝑥 , then solve using the properties of 𝑠𝑖𝑛𝑥 and 𝑐𝑜𝑠𝑥 . (2) Change into sec2x, as derivative of tan x is sec2 . Here, 1st Method is not applicable , so we have used 2nd Method . = √(tan𝑥 )/(cos^2𝑥 . tan𝑥 ) = (tan𝑥 )^(1/2 − 1) × 1/cos^2𝑥 = (tan𝑥 )^((−1)/2) × 1/cos^2𝑥 = (tan𝑥 )^((−1)/2) × sec^2𝑥 ∴ √(tan𝑥 )/sin〖𝑥 cos𝑥 〗 " = " (tan𝑥 )^((−1)/2) " × " sec^2𝑥 Step 2: Integrating the function ∫1▒〖 √(tan𝑥 )/sin〖𝑥 cos𝑥 〗 〗 . 𝑑𝑥 = ∫1▒〖 (tan𝑥 )^((−1)/2) " × " sec^2𝑥 〗. 𝑑𝑥" " Let tan𝑥 = 𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 sec^2𝑥=𝑑𝑡/𝑑𝑥 𝑑𝑥=𝑑𝑡/sec^2𝑥 Thus, our equation becomes ∴ ∫1▒〖 (tan𝑥 )^((−1)/2) " ." sec^2𝑥 〗. 𝑑𝑥" " = ∫1▒〖 (𝑡)^((−1)/2) " " . sec^2𝑥 〗. 𝑑𝑡/sec^2𝑥 " " = ∫1▒〖𝑡^((−1)/2) . 𝑑𝑡〗 = 𝑡^(− 1/2 +1)/(− 1/2 +1) + 𝐶 = 𝑡^(1/2)/(1/2) + 𝐶 = 〖2𝑡〗^(1/2)+ 𝐶 = 2√𝑡+ 𝐶 = 𝟐√(𝐭𝐚𝐧𝒙 )+ 𝑪 (Using 𝑡=𝑡𝑎𝑛𝑥)