CBSE Class 12 Sample Paper for 2025 Boards
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CBSE Class 12 Sample Paper for 2025 Boards
Last updated at Dec. 13, 2024 by Teachoo
This question is similar to Chapter 7 Class 12 Integrals - Miscellaneous
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Question 12 ∫𝑑𝑥/(𝑥^3 (1 + 𝑥^4 )^(1/2) ) equals (A) −1/(2𝑥^2 ) √(1+𝑥^4 )+𝑐 (B) 1/2𝑥 √(1+𝑥^4 )+𝑐 (C) −1/4𝑥 √(1+𝑥^4 )+𝑐 (D) 1/(4𝑥^2 ) √(1+𝑥^4 )+𝑐∫1▒( 𝑑𝑥)/(𝑥^3 (1 + 𝑥^4 )^(1/2) ) Taking 𝒙^𝟒 common from denominator = ∫1▒𝑑𝑥/(𝑥^(3 ) 〖〖(𝑥〗^4)〗^(1/2) ( 1/𝑥^4 + 1)^(1/2) ) = ∫1▒𝑑𝑥/(𝑥^3 〖(𝑥〗^2) (1 + 1/𝑥^4 )^(1/2) ) = ∫1▒𝒅𝒙/(𝒙^𝟓 (𝟏 + 𝟏/𝒙^𝟒 )^(𝟏/𝟐) ) Let t = 1 + 𝟏/𝒙^𝟒 𝑑𝑡/𝑑𝑥 = −4/𝑥^5 −𝑑𝑡/4 = 𝑑𝑥/𝑥^5 Substituting value of x and dx = (−1)/4 ∫1▒𝑑𝑡/𝑡^(1/2) = (−1)/4 ∫1▒𝑡^((−1)/2) 𝑑𝑡 = (−1)/4 [𝑡^((−1)/2 + 1)/((−1)/2 + 1)]+ C = (−1)/4 (𝑡^(1/2)/(1/2))+ C = (−1)/2 𝑡^(1/2)+ C = (−1)/2 (1+1/𝑥^4 )^(1/2)+ C = (−𝟏)/𝟐 √(𝟏+𝟏/𝒙^𝟒 )+ C = (−1)/2 √((𝑥^4 + 1)/𝑥^4 )+ C = (−1)/2 ×1/√(𝑥^4 ) √(𝑥^4 + 1)+ C = (−𝟏)/(𝟐𝒙^𝟐 ) √(𝒙^𝟒 + 𝟏)+ C So, the correct answer is (A)
