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Example 12 - Find x2 + 1 / x2 - 5x + 6 dx - Class 12 - Integration by partial fraction - Type 1

Example 12 - Chapter 7 Class 12 Integrals - Part 2
Example 12 - Chapter 7 Class 12 Integrals - Part 3

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Transcript

Example 12 Find ﷮﷮ 𝑥﷮2﷯+ 1﷮ 𝑥﷮2﷯ − 5𝑥 + 6﷯﷯ 𝑑𝑥 Solving Integrand 𝑥﷮2﷯+ 1﷮ 𝑥﷮2﷯ − 5𝑥 + 6﷯ 𝑥﷮2﷯+ 1﷮ 𝑥﷮2﷯ − 5𝑥 + 6﷯= 𝑥﷮2﷯+ 1 − 5𝑥 + 6 + 5𝑥 − 6﷮ 𝑥﷮2﷯ − 5𝑥 + 6﷯ = 𝑥﷮2﷯ − 5𝑥 + 6﷯ + 5𝑥 − 5﷯﷮ 𝑥﷮2﷯ − 5𝑥 + 6﷯ = 𝑥﷮2﷯ − 5𝑥 + 6﷮ 𝑥﷮2﷯ − 5𝑥 + 6﷯ + 5𝑥 − 5﷮ 𝑥﷮2﷯ − 5𝑥 + 6﷯ =1+ 5𝑥 − 5﷮ 𝑥﷮2﷯ − 5𝑥 + 6﷯ =1+ 5𝑥 − 5﷮ 𝑥﷮2﷯ − 2𝑥 − 3𝑥 + 6﷯ =1+ 5𝑥 − 5﷮𝑥 𝑥 − 2﷯ −3 𝑥 − 2﷯﷯ =1+ 5𝑥 − 5﷮ 𝑥 − 2﷯ 𝑥 − 3﷯﷯ Now solving 5𝑥 − 5﷮ 𝑥 − 2﷯ 𝑥 − 3﷯﷯ using partial fractions Let 5 𝑥 − 1﷯﷮ 𝑥 − 2﷯ 𝑥 − 3﷯﷯= 𝐴﷮ 𝑥 − 2﷯﷯ + 𝐵﷮ 𝑥 − 3﷯﷯ 5 𝑥 − 1﷯﷮ 𝑥 − 2﷯ 𝑥 − 3﷯﷯= 𝐴 𝑥 − 3﷯ + 𝐵 𝑥 − 2﷯﷮ 𝑥 − 2﷯ 𝑥 − 3﷯﷯ ⇒5 𝑥−1﷯=𝐴 𝑥−3﷯+𝐵 𝑥−2﷯ Hence we can write it as 5 𝑥 − 1﷯﷮ 𝑥 − 2﷯ 𝑥 − 3﷯﷯ = 𝐴﷮ 𝑥 − 2﷯﷯ + 𝐵﷮ 𝑥 − 3﷯﷯ Now, ﷮﷮ 𝑥﷮2﷯+ 1﷮ 𝑥﷮2﷯ − 5𝑥 + 6﷯﷯ 𝑑𝑥= ﷮﷮ 1+ 5 𝑥 − 1﷯﷮ 𝑥 − 2﷯ 𝑥 − 3﷯﷯﷯﷯ 𝑑𝑥 = ﷮﷮1+ −5﷮ 𝑥 − 2﷯﷯ + 10﷮ 𝑥 − 3﷯﷯﷯ 𝑑𝑥 = ﷮﷮1﷯𝑑𝑥− ﷮﷮ −5﷮ 𝑥 − 2﷯﷯﷯ 𝑑𝑥+ ﷮﷮ 10﷮ 𝑥 − 3﷯﷯﷯ 𝑑𝑥 =𝒙−𝟓 𝒍𝒐𝒈﷮ 𝒙−𝟐﷯﷯+𝟏𝟎 𝒍𝒐𝒈﷮ 𝒙−𝟑﷯﷯+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.