Ex 7.9, 8 - Chapter 7 Class 12 Integrals (Important Question)

Last updated at Dec. 16, 2024 by Teachoo

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Ex 7.9, 8 Evaluate the integrals using substitution ∫_1^(2 )▒〖 (1/𝑥 −1/(2𝑥^2 )) 〗 𝑒^2𝑥 𝑑𝑥 Let 𝑡=2𝑥 𝑑𝑡/𝑑𝑥=2 𝑑𝑡/2=𝑑𝑥 Thus, when x varies from 1 to 2, t varies from 2 to 4 Substituting, ∫_1^(2 )▒〖 (1/𝑥 −1/(2𝑥^2 )) 〗 𝑒^2𝑥 𝑑𝑥 = ∫_2^4▒〖𝑒^𝑡 (1/(𝑡/2)−1/(2〖 (𝑡/2)〗^2 )) 〗 𝑑𝑡/2 =∫_2^4▒〖𝑒^𝑡 (2/𝑡−4/(2𝑡^2 )) 〗 𝑑𝑡/2 =∫_2^4▒〖𝑒^𝑡 (1/𝑡−2/𝑡^2 ) 〗 𝑑𝑡 It is of the form ∫1▒〖𝑒^𝑥 [𝑓(𝑥)+𝑓^′ (𝑥)] 〗 𝑑𝑥=𝑒^𝑥 𝑓(𝑥)+𝐶 Where 𝑓(𝑥)=1/𝑡 𝑓^′ (𝑥)= (−1)/𝑡^2 Hence, our equation becomes ∫_2^4▒〖𝑒^𝑡 (1/𝑡−2/𝑡^2 ) 〗 𝑑𝑡 = [𝑒^𝑡×1/𝑡]_2^4 = (𝑒^4/4−𝑒^2/2) = (𝒆^𝟐 (𝒆^𝟐 − 𝟐))/𝟒

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo

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