Question 20

Evaluate: Definite Integration -1 → 1  ∫ (x + |x| + 1) / (x + 2|x| + 1)

Find Integration  (x + |x| + 1) / (x2 + 2|x| + 1) from -1 to 1

Question 20 - CBSE Class 12 Sample Paper for 2019 Boards - Part 2
Question 20 - CBSE Class 12 Sample Paper for 2019 Boards - Part 3
Question 20 - CBSE Class 12 Sample Paper for 2019 Boards - Part 4
Question 20 - CBSE Class 12 Sample Paper for 2019 Boards - Part 5


Transcript

Question 20 Evaluate: โˆซ1_(โˆ’1)^1โ–’(๐‘ฅ+|๐‘ฅ|+1)/(๐‘ฅ^2+2|๐‘ฅ|+1) We know that |๐‘ฅ|={โ–ˆ(โˆ’&๐‘ฅ, ๐‘ฅ<0@&๐‘ฅ, ๐‘ฅโ‰ฅ0)โ”ค So, for โ€“1 to 0, |x| = โ€“x and for 0 to 1 |x| = x So, our integral becomes โˆซ1_(โˆ’1)^1โ–’(๐‘ฅ+|๐‘ฅ|+1)/(๐‘ฅ^2+2|๐‘ฅ|+1) โˆซ1_(โˆ’1)^0โ–’(๐‘ฅ+|๐‘ฅ|+1)/(๐‘ฅ^2+2|๐‘ฅ|+1) โˆซ1_0^1โ–’(๐‘ฅ+|๐‘ฅ|+1)/(๐‘ฅ^2+2|๐‘ฅ|+1) โˆซ1_(โˆ’1)^0โ–’(๐‘ฅโˆ’๐‘ฅ+1)/(๐‘ฅ^2โˆ’2๐‘ฅ+1) โˆซ1_0^1โ–’(๐‘ฅ+๐‘ฅ+1)/(๐‘ฅ^2+2๐‘ฅ+1) โˆซ1_(โˆ’1)^0โ–’1/(๐‘ฅ^2โˆ’2๐‘ฅ+1) โˆซ1_0^1โ–’(2๐‘ฅ+1)/(๐‘ฅ^2+2๐‘ฅ+1) โˆซ1_(โˆ’1)^0โ–’1/(๐‘ฅโˆ’1)^2 โˆซ1_0^1โ–’(2๐‘ฅ+1)/(๐‘ฅ+1)^2 โˆซ1_(โˆ’1)^0โ–’1/(๐‘ฅโˆ’1)^2 โˆซ1_0^1โ–’(2๐‘ฅ+1+1โˆ’1)/(๐‘ฅ+1)^2 โˆซ1_(โˆ’1)^0โ–’1/(๐‘ฅโˆ’1)^2 โˆซ1_0^1โ–’(2๐‘ฅ+2โˆ’1)/(๐‘ฅ+1)^2 โˆซ1_(โˆ’1)^0โ–’1/(๐‘ฅโˆ’1)^2 โˆซ1_0^1โ–’(2(๐‘ฅ+1)โˆ’1)/(๐‘ฅ+1)^2 โˆซ1_(โˆ’1)^0โ–’1/(๐‘ฅโˆ’1)^2 โˆซ1_0^1โ–’(2(๐‘ฅ+1) )/(๐‘ฅ+1)^2 โˆซ1_0^1โ–’1/(๐‘ฅ+1)^2 โˆซ1_(โˆ’1)^0โ–’1/(๐‘ฅโˆ’1)^2 โˆซ1_0^1โ–’(2 )/((๐‘ฅ+1) ) โˆซ1_0^1โ–’1/(๐‘ฅ+1)^2 โˆซ1_(โˆ’1)^0โ–’1/(๐‘ฅโˆ’1)^2 2 โˆซ1_0^1โ–’"dx " /((๐‘ฅ+1) ) โˆซ1_0^1โ–’1/(๐‘ฅ+1)^2 [(๐‘ฅโˆ’1)^(โˆ’1)/(โˆ’1)]_(โˆ’1)^0 2[lnโกใ€–|๐‘ฅ+1|ใ€— ]_0^1 [(๐‘ฅ+1)^(โˆ’1)/(โˆ’1)]_0^1 [(โˆ’1)/((๐‘ฅโˆ’1))]_(โˆ’1)^0 "2" [lnโกใ€–|๐‘ฅ+1|ใ€— ]_0^1 [(โˆ’1)/((๐‘ฅ+1))]_0^1 [(โˆ’1)/((0โˆ’1))โˆ’(โˆ’1)/((โˆ’1โˆ’1))] 2[lnโกใ€–|2|ใ€—โˆ’logโกใ€–|1|ใ€— ] [(โˆ’1)/((1+1))โˆ’(โˆ’1)/((0+1))] [1โˆ’1/2] 2[lnโกใ€–|2|ใ€—โˆ’0] [โˆ’1/2+1] 1/2+2 lnโก2โˆ’1/2 2 lnโก2

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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, Social Science, Physics, Chemistry, Computer Science at Teachoo.