Ex 7.5, 21 - Chapter 7 Class 12 Integrals (Important Question)
Last updated at Dec. 16, 2024 by Teachoo
Chapter 7 Class 12 Integrals
Ex 7.1, 18 Important
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Ex 7.2, 35
Ex 7.2, 36 Important
Ex 7.3, 6 Important
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Ex 7.5, 18 Important
Ex 7.5, 21 Important You are here
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Ex 7.6, 19
Ex 7.6, 24 (MCQ) Important
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Ex 7.7, 11 Important
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Example 25 (i)
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Ex 7.8, 16 Important
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Misc 18 Important
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Question 1 Important
Misc 23 Important
Misc 29 Important
Question 2 Important
Misc 38 (MCQ) Important
Question 4 (MCQ) Important
Integration Formula Sheet Important
Chapter 7 Class 12 Integrals
Last updated at Dec. 16, 2024 by Teachoo
Ex 7.5, 21 Integrate the function 1/((๐^๐ฅ โ 1) ) [Hint : Put ex = t] Let ๐^๐ฅ = ๐ก Differentiating both sides ๐ค.๐.๐ก.๐ฅ ๐^๐ฅ = ๐๐ก/๐๐ฅ ๐๐ฅ = ๐๐ก/๐^๐ฅ Therefore โซ1โ1/((๐^๐ฅ โ 1) ) ๐๐ฅ = โซ1โ1/((๐ก โ 1) ) ๐๐ก/๐^๐ฅ = โซ1โ๐๐ก/(๐ก(๐ก โ 1) ) We can write integrand as 1/(๐ก(๐ก โ 1) ) = ๐ด/๐ก + ๐ต/(๐ก โ 1) 1/(๐ก(๐ก โ 1) ) = (๐ด(๐ก โ 1) + ๐ต๐ก)/๐ก(๐ก โ 1) Cancelling denominator 1 = ๐ด(๐กโ1)+๐ต๐ก Putting t = 0 in (1) 1 = ๐ด(0โ1)+๐ตร0 1 = ๐ดร(โ1) 1 = โ๐ด ๐ด = โ1 Putting t = 1 1 = ๐ด(1โ1)+๐ตร1 1 = ๐ดร0+๐ต 1 = ๐ต ๐ต = 1 Therefore โซ1โ1/(๐ก(๐ก โ 1) ) ๐๐ก = โซ1โ(โ1)/(๐ก ) ๐๐ก + โซ1โ1/(๐ก โ 1 ) = โใlog ใโก|๐ก|+ใlog ใโก|๐กโ1|+๐ถ = ใlog ใโก|(๐ก โ 1)/๐ก|+๐ถ Putting back t = ๐^๐ฅ = ใ๐๐๐ ใโก|(๐^๐ฅ โ 1)/๐^๐ฅ |+๐ถ = ใ๐ฅ๐จ๐ ใโก((๐^๐ โ ๐)/๐^๐ )+๐ช Since ex > 1 for x > 0 โด ex โ 1 > 0 โ |(๐^๐ โ ๐)/๐^๐ |=((๐^๐ โ ๐)/๐^๐ ) ("As " ๐๐๐ ๐ดโ๐๐๐ ๐ต" = " ๐๐๐ ๐ด/๐ต)