Chapter 7 Class 12 Integrals
Concept wise

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Ex 7.9, 4 Evaluate the integrals using substitution โˆซ_0^2โ–’ใ€–๐‘ฅโˆš(๐‘ฅ+2)ใ€—โกใ€– (๐‘๐‘ข๐‘ก ๐‘ฅ+2=๐‘ก^2 )ใ€— โˆซ_0^2โ–’ใ€–๐‘ฅโˆš(๐‘ฅ+2)ใ€—โกใ€– ๐‘‘๐‘ฅใ€— Put ๐‘ฅ+2=๐‘ก^2 Differentiating w.r.t. ๐‘ฅ ๐‘‘(๐‘ฅ + 2)/๐‘‘๐‘ฅ=๐‘‘(๐‘ก^2 )/๐‘‘๐‘ก ร—๐‘‘๐‘ก/๐‘‘๐‘ฅ 1=2๐‘ก ร— ๐‘‘๐‘ก/๐‘‘๐‘ฅ ๐‘‘๐‘ฅ=2๐‘ก ๐‘‘๐‘ก Hence, when ๐‘ฅ varies from 0 to 2, then t varies from โˆš2 to 2 Therefore we can write โˆซ_0^2โ–’ใ€–๐‘ฅโˆš(๐‘ฅ+2) ๐‘‘๐‘ฅ =โˆซ_(โˆš2)^2โ–’ใ€–(๐‘ก^2โˆ’2) โˆš(๐‘ก^2 ) 2๐‘ก ๐‘‘๐‘กใ€—ใ€— =โˆซ_(โˆš2)^2โ–’ใ€–(๐‘ก^2โˆ’2)๐‘ก ร—2๐‘ก ๐‘‘๐‘กใ€— =2โˆซ_(โˆš2)^2โ–’ใ€–(๐‘ก^2โˆ’2) ๐‘ก^2 ๐‘‘๐‘กใ€— =2โˆซ_(โˆš2)^2โ–’ใ€–(๐‘ก^4โˆ’2๐‘ก^2 ) ๐‘‘๐‘กใ€— =2[๐‘ก^(4+1)/(4+1)โˆ’2 ๐‘ก^(2+1)/(2+1)]_(โˆš2)^2 =2[๐‘ก^5/5โˆ’2 ๐‘ก^3/3]_(โˆš2)^2 =2ร— [2^5/5โˆ’2/3 2^3โˆ’((โˆš2)^5/5โˆ’2/3 (โˆš2)^3 )] =2ร— [๐‘ฅ^5/2โˆ’2/3 2^3โˆ’((โˆš2)^5/2โˆ’2/3 (โˆš2)^3 )] =2ร— [32/5โˆ’16/3โˆ’(4โˆš2)/5+4/3 โˆš2] =2ร— [(96 โˆ’ 80 โˆ’ 12โˆš2 + 20โˆš2)/15] =2ร— [(16 + 8โˆš2)/15] =(2 ร— 8 โˆš2 (โˆš2 + 1))/15 =(๐Ÿ๐Ÿ”โˆš(๐Ÿ ) (โˆš๐Ÿ + ๐Ÿ))/๐Ÿ๐Ÿ“

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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.