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Ex 7.10, 1 Evaluate the integrals using substitution ∫_0^1▒〖𝑥/(𝑥^2 + 1) 𝑑𝑥〗 We need to find ∫_𝟎^𝟏▒〖𝒙/(𝒙^𝟐 + 𝟏) 𝒅𝒙〗 Let 𝒕=𝒙^𝟐+𝟏 Differentiating w.r.t. 𝑥 𝑑𝑡/𝑑𝑥=𝑑/𝑑𝑥 (𝑥^2+1) 𝑑𝑡/𝑑𝑥=2𝑥 𝒅𝒕/𝟐𝒙=𝒅𝒙 Now, when 𝒙 varies from 0 to 1 then 𝒕 varies from 1 to 2 Therefore ∫_𝟎^𝟏▒〖𝒙/(𝒙^𝟐+𝟏) 𝒅𝒙=∫_𝟏^𝟐▒〖𝒙/𝒕 𝒅𝒕/𝟐𝒙〗〗 =1/2 ∫_1^2▒𝑑𝑡/𝑡 =𝟏/𝟐 [𝒍𝒐𝒈|𝒕|]_𝟏^𝟐 =1/2 [𝑙𝑜𝑔|2|−𝑙𝑜𝑔|1|] =1/2 [𝑙𝑜𝑔|2|−0] =1/2 𝑙𝑜𝑔|2| =𝟏/𝟐 𝒍𝒐𝒈 𝟐

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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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.