Ex 7.9, 8 - Chapter 7 Class 12 Integrals

Last updated at Dec. 16, 2024 by Teachoo

Next: Misc 33 → Go Ad-free

Transcript

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

Made by

Davneet Singh

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

Class 6

Class 7

Class 8

Class 9

Class 10

Class 11

Class 12

Courses

Interesting

Popular