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 Ex 7.10, 4 - Evaluate integrals x root x + 2 (put x + 2 = t2) - Definate Integration - By Substitution

Ex 7.10, 4 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.10, 4 - Chapter 7 Class 12 Integrals - Part 3

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Transcript

Ex 7.10, 4 Evaluate the integrals using substitution 0﷮2﷮ 𝑥 ﷮𝑥+2﷯﷮ 𝑝𝑢𝑡 𝑥+2= 𝑡﷮2﷯﷯﷯﷯ 0﷮2﷮ 𝑥 ﷮𝑥+2﷯﷮ 𝑑𝑥﷯﷯ Put 𝑥+2= 𝑡﷮2﷯ Differentiating w.r.t. 𝑥 𝑑 𝑥 + 2﷯﷮𝑑𝑥﷯= 𝑑 𝑡﷮2﷯﷯﷮𝑑𝑡﷯ × 𝑑𝑡﷮𝑑𝑥﷯ 1=2𝑡 × 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥=2𝑡 𝑑𝑡 Hence, when 𝑥 varies from 0 to 2, then t varies from ﷮2﷯ to 2 Therefore we can write 0﷮2﷮𝑥 ﷮𝑥+2﷯ 𝑑𝑥 = ﷮2﷯﷮2﷮ 𝑡﷮2﷯−2﷯ ﷮ 𝑡﷮2﷯﷯ 2𝑡 𝑑𝑡﷯﷯ = ﷮2﷯﷮2﷮ 𝑡﷮2﷯−2﷯𝑡 ×2𝑡 𝑑𝑡﷯ =2 ﷮2﷯﷮2﷮ 𝑡﷮2﷯−2﷯ 𝑡﷮2﷯ 𝑑𝑡﷯ =2 ﷮2﷯﷮2﷮ 𝑡﷮4﷯−2 𝑡﷮2﷯﷯ 𝑑𝑡﷯ =2 𝑡﷮4+1﷯﷮4+1﷯−2 𝑡﷮2+1﷯﷮2+1﷯﷯﷮ ﷮2﷯﷮2﷯ =2 𝑡﷮5﷯﷮5﷯−2 𝑡﷮3﷯﷮3﷯﷯﷮ ﷮2﷯﷮2﷯ =2× 2﷮5﷯﷮5﷯− 2﷮3﷯ 2﷮3﷯− ﷮2﷯﷯﷮5﷯﷮5﷯− 2﷮3﷯ ﷮2﷯﷯﷮3﷯﷯﷯ =2× 𝑥﷮5﷯﷮2﷯− 2﷮3﷯ 2﷮3﷯− ﷮2﷯﷯﷮5﷯﷮2﷯− 2﷮3﷯ ﷮2﷯﷯﷮3﷯﷯﷯ =2× 32﷮5﷯− 16﷮3﷯− 4 ﷮2﷯﷮5﷯+ 4﷮3﷯ ﷮2﷯﷯ =2× 96 − 80 − 12 ﷮2﷯ + 20 ﷮2﷯﷮15﷯﷯ =2× 16 + 8 ﷮2﷯﷮15﷯﷯ = 2 × 8 ﷮2﷯ ﷮2﷯ + 1﷯﷮15﷯ = 𝟏𝟔 ﷮𝟐 ﷯ ﷮𝟐﷯ + 𝟏﷯﷮𝟏𝟓﷯

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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.