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Transcript

Ex 7.8, 22 Choose the correct answer โˆซ_0^(2/3)โ–’๐‘‘๐‘ฅ/(4 +9๐‘ฅ^2 ) equals (A) ๐œ‹/6 (B) ๐œ‹/12 (C) ๐œ‹/24 (D) ๐œ‹/4 Step 1 :- Let F(๐‘ฅ)=โˆซ1โ–’๐‘‘๐‘ฅ/(4 + 9๐‘ฅ^2 ) Divide and multiply the integrate by 4 =โˆซ1โ–’โ–ˆ(๐‘‘๐‘ฅ@4)/((4 + 9๐‘ฅ^2)/4) =1/4 โˆซ1โ–’๐‘‘๐‘ฅ/(1 + 9/4 ๐‘ฅ^2 ) =1/4 โˆซ1โ–’๐‘‘๐‘ฅ/(1 + (3/2 ๐‘ฅ)^2 ) Put 3/2 ๐‘ฅ=๐‘ก Differentiating w.r.t.๐‘ฅ ๐‘‘(3/2 ๐‘ฅ)/๐‘‘๐‘ฅ=๐‘‘๐‘ก/๐‘‘๐‘ฅ 3/2=๐‘‘๐‘ก/๐‘‘๐‘ฅ ๐‘‘๐‘ฅ=๐‘‘๐‘ก/(3/2) ๐‘‘๐‘ฅ=2/3 ๐‘‘๐‘ก Hence 1/4 โˆซ1โ–’ใ€–๐‘‘๐‘ฅ/(1+(3/2 ๐‘ฅ)^2 )=1/4 โˆซ1โ–’ใ€–1/(1+๐‘ก^2 ) 2/3ใ€— ๐‘‘๐‘กใ€— =1/4 ร— 2/3 โˆซ1โ–’๐‘‘๐‘ก/(1+๐‘ก^2 ) =1/6 tan^(โˆ’1)โก๐‘ก Putting ๐‘ก=3/2 ๐‘ฅ =1/6 tan^(โˆ’1)โกใ€–3/2 ๐‘ฅใ€— Hence F(๐‘ฅ)=1/6 tan^(โˆ’1)โกใ€–3/2 ๐‘ฅใ€— Step 2 :- โˆซ_0^(2/3)โ–’ใ€–๐‘‘๐‘ฅ/(4+9๐‘ฅ^2 )=๐น(2/3)โˆ’๐น(0) ใ€— =1/6 tan^(โˆ’1)โกใ€–(3/2 ร— 2/3)โˆ’1/6 tan^(โˆ’1)โก(0) ใ€— =1/6 tan^(โˆ’1)โกใ€–(1)โˆ’1/6 tan^(โˆ’1)โก(0) ใ€— =1/6 ร— ๐œ‹/4โˆ’0 =๐œ‹/24 โˆต Option (C) is correct.

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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.