Example 16 - Chapter 7 Class 12 Integrals (Term 2)
Last updated at Dec. 20, 2019 by Teachoo
Transcript
Example 16 Find โซ1โ(๐ฅ^2+ ๐ฅ +1 ๐๐ฅ )/((๐ฅ + 2) (๐ฅ^2+1) ) We can write equation as (๐ฅ^2+ ๐ฅ + 1)/((๐ฅ + 1) (๐ฅ + 2) )=๐ด/(๐ฅ + 2) + (๐ต๐ฅ + ๐ถ)/(๐ฅ^2+ 1) Cancelling denominator ใ ๐ฅใ^2+ ๐ฅ+1=๐ด(๐ฅ^2+1)+(๐ต๐ฅ+๐ถ) (๐ฅ+2) Putting x = โ๐ (โ2)^2+(โ2)+1=๐ด((โ2)^2+1)+0 4โ2+1= 5A 3/5 = A Putting x = ๐ ๐ฅ^2+ ๐ฅ+1=๐ด(๐ฅ^2+1)+(๐ต๐ฅ+๐ถ) (๐ฅ+2) 0+0+0= A(0 + 1) + (0 + C) (0 + 2) 1 = A + 2C 1 = 3/5 + 2C 1 โ 3/5 = 2C 2/5 = 2C C = 1/5 Putting x = 1 ๐ฅ^2+ ๐ฅ+1=๐ด(๐ฅ^2+1)+(๐ต๐ฅ+๐ถ) (๐ฅ+2) 1+1+1= 2A + (B + C)(3) 3 = 2A + 3 (B + C) 3 = 2(3/5) + 3 (B+1/5) 3 โ 6/5 = 3 (B+1/5) 9/5 = 3 (B+1/5) 3/5 โ 1/5 = B B = 2/5 Thus, (๐ฅ^2+ ๐ฅ + 1)/((๐ฅ + 1) (๐ฅ + 2) )=๐ด/(๐ฅ + 2) + (๐ต๐ฅ + ๐ถ)/(๐ฅ^2+ 1) (๐ฅ^2+ ๐ฅ + 1)/((๐ฅ + 1)(๐ฅ^2+ 1)) = 3/(5 (๐ฅ + 2)) + (1 (2๐ฅ + 1))/(5 (๐ฅ^2 + 1)) Hence, our equation becomes โซ1โ(๐ฅ^2+ ๐ฅ + 1)/((๐ฅ + 2) (๐ฅ^2 + 1)) ๐๐ฅ= โซ1โ3/(5(๐ฅ^2 + 1)) ๐๐ฅ+โซ1โ1/5 ((2๐ฅ + 1))/(๐ฅ^2 + 1) ๐๐ฅ = โซ1โ3/(5(๐ฅ^2 + 1)) ๐๐ฅ+ 1/5 โซ1โใ2๐ฅ/(๐ฅ^2 + 1) ๐๐ฅ+ใ 1/5 โซ1โ1/(๐ฅ^2 + 1) ๐๐ฅ ๐๐ 1/5 โซ1โ2๐ฅ/(๐ฅ^2+ 1) ๐๐ฅ Let ๐ก=๐ฅ^2+ 1 ๐๐ก/๐๐ฅ=2๐ฅ ๐๐ก=2๐ฅ ๐๐ฅ Substituting, =1/5 โซ1โ๐๐ก/๐ก = 1/5 log |๐ก| + C_2 = 1/5 log |๐ฅ^2+1| + C_2 ๐๐ 1/5 โซ1โ1/(๐ฅ^2+ 1) ๐๐ฅ = 1/5 ใ๐ก๐๐ใ^(โ1) (๐ฅ)+C_3 Hence โซ1โ(๐ฅ^2+ ๐ฅ + 1)/((๐ฅ + 2) (๐ฅ^2+ 1)) ๐๐ฅ =๐/๐ ๐๐๐|๐+๐|+๐/๐ ๐๐๐|๐^๐+๐|+๐/๐ ใ๐๐๐ใ^(โ๐) (๐)+ C where C = C_1+ C_2+C_3
Examples
Example 1 (i)
Example 1 (ii)
Example 1 (iii)
Example 2 (i)
Example 2 (ii)
Example 2 (iii) Important
Example 3 (i)
Example 3 (ii) Important
Example 3 (iii)
Example 4
Example 5 (i)
Example 5 (ii)
Example 5 (iii) Important
Example 5 (iv) Important
Example 6 (i)
Example 6 (ii) Important
Example 6 (iii) Important
Example 7 (i)
Example 7 (ii) Important
Example 7 (iii)
Example 8 (i)
Example 8 (ii) Important
Example 9 (i)
Example 9 (ii) Important
Example 9 (iii) Important
Example 10 (i)
Example 10 (ii) Important
Example 11
Example 12
Example 13 Important
Example 14
Example 15 Important
Example 16 Important You are here
Example 17
Example 18 Important
Example 19
Example 20 Important
Example 21 Important
Example 22 Important
Example 23
Example 24
Example 25 Important Deleted for CBSE Board 2022 Exams
Example 26 Deleted for CBSE Board 2022 Exams
Example 27 (i)
Example 27 (ii) Important
Example 27 (iii)
Example 27 (iv) Important
Example 28
Example 29
Example 30 Important
Example 31
Example 32
Example 33
Example 34 Important
Example 35 Important
Example 36 Important
Example 37
Example 38 Important
Example 39 Important
Example 40 Important
Example 41 Important
Example 42 Important
Example 43 Important
Example 44 Important
Example 25 (Supplementary NCERT) Important Deleted for CBSE Board 2022 Exams
About the Author
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.