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Ex 7.5, 5 - Integrate 2x / x2 + 3x + 2 - Chapter 7 Class 12 - Ex 7.5

Ex 7.5, 5 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.5, 5 - Chapter 7 Class 12 Integrals - Part 3

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Transcript

Ex 7.5, 5 2𝑥﷮ 𝑥﷮2﷯ + 3𝑥 + 2﷯ Solving integrand 2𝑥﷮ 𝑥﷮2﷯ + 3𝑥 + 2﷯ = 2𝑥﷮ 𝑥﷮2﷯ + 2𝑥 + 𝑥 + 2﷯ = 2𝑥﷮𝑥 𝑥 + 2﷯ + 1 𝑥 + 2﷯﷯ = 2𝑥﷮ 𝑥 + 2﷯ 𝑥 + 1﷯﷯ We can write it as 2𝑥﷮ 𝑥 + 2﷯ 𝑥 + 1﷯﷯ = 𝐴﷮ 𝑥 + 2﷯﷯ + 𝐵﷮ 𝑥 + 1﷯﷯ 2𝑥﷮ 𝑥 + 2﷯ 𝑥 + 1﷯﷯ = 𝐴 𝑥 + 1﷯ + 𝐵 𝑥 + 2﷯﷮ 𝑥 + 2﷯ 𝑥 + 1﷯﷯ Cancelling denominator 2𝑥 = 𝐴 𝑥+1﷯+𝐵 𝑥+2﷯ Putting x = −1, in (1) 2 −1﷯ = 𝐴 −1+1﷯ + 𝐵 −1+2﷯ −2 = 𝐴×0+𝐵 1﷯ 𝐵 = −2 Similarly putting x = − 2 , in (1) 2 −2﷯ = 𝐴 −2+1﷯ + 𝐵 −2+2﷯ −4 = 𝐴 −1﷯+𝐵×0 −4 = −𝐴 𝐴 = 4 Hence we can write it as 2𝑥﷮ 𝑥 + 2﷯ 𝑥 + 1﷯﷯ = 4﷮ 𝑥 + 2﷯﷯ + −2﷯﷮ 𝑥 + 1﷯﷯ = 4﷮ 𝑥 + 2﷯﷯ − 2﷮ 𝑥 + 1﷯﷯ Integrating 𝑤.𝑟.𝑡.𝑥. ﷮﷮ 2𝑥﷮ 𝑥 + 2﷯ 𝑥 + 1﷯﷯﷯ 𝑑𝑥 = ﷮﷮ 4﷮ 𝑥 + 2﷯﷯ − 2﷮ 𝑥 + 1﷯﷯﷯﷯𝑑𝑥 = 4 ﷮﷮ 𝑑𝑥﷮ 𝑥 + 2﷯﷯﷯−2 ﷮﷮ 𝑑𝑥﷮ 𝑥 + 1﷯﷯ ﷯ = 𝟒 𝐥𝐨𝐠 ﷮ 𝒙+𝟐﷯﷯−𝟐 𝐥𝐨𝐠 ﷮ 𝒙+𝟏﷯﷯ +𝑪

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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.