Example 13 - Find integral 3x - 2 / (x + 1)2 (x + 3) dx - Examples

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Example 13 Find ﷮﷮ 3𝑥 −2﷮ 𝑥 + 1﷯﷮2﷯ 𝑥 + 3﷯﷯﷯ 𝑑𝑥 We can write Integral as 3𝑥 − 2﷮ 𝑥 + 1﷯﷮2﷯ 𝑥 + 3﷯﷯= 𝐴﷮𝑥 + 1﷯ + 𝐵﷮ 𝑥 + 1﷯﷮2﷯﷯ + 𝐵﷮ 𝑥 + 3﷯﷯ 3𝑥 − 2﷮ 𝑥 + 1﷯﷮2﷯ 𝑥 + 3﷯﷯= 𝐴 𝑥 + 1﷯ 𝑥 + 3﷯ + 𝐵 𝑥 + 3﷯ + 𝐶 𝑥 + 1﷯﷮2﷯﷮ 𝑥 + 1﷯﷮2﷯ 𝑥 + 3﷯﷯ By cancelling denominator 3𝑥 −2=𝐴 𝑥+1﷯ 𝑥+3﷯+𝐵 𝑥+3﷯+𝐶 𝑥+1﷯﷮2﷯ Putting x = −1 3 −1﷯ −2=𝐴 −1+1﷯ −1+3﷯+𝐵 −1+3﷯+𝐶 −1+1﷯﷮2﷯ −3−2=𝐴×0+𝐵×2+𝐶× 0﷯﷮2﷯ −5=𝐵×2 𝐵= − 5﷮2﷯ Similarly putting x = − 3 3 −3﷯−2=𝐴 −3+1﷯ −3+3﷯+𝐵 −3+3﷯+𝐶 −3+1﷯﷮2﷯ −9−2=𝐴×0+𝐵×0+𝐶× −2﷯﷮2﷯ −11=0+0+𝐶 4﷯ −11=4𝐶 −11﷮4﷯ =𝐶 𝐶 = −11﷮4﷯ Now, Putting x = 0 3(0) − 2 = A(1) (3) + B(3) + C (1)﷮2﷯ −2 = 3A + 3B + C Putting value of B & C −2 = 3A + 3 −5﷮2﷯﷯ + −11﷮4﷯﷯ −2 = 3A − 15﷮2﷯− 11﷮4﷯ −2 = 3A + −30 − 11﷮4﷯ −8 = 12A − 41 41 − 8 = 12A 33﷮12﷯ = A 11﷮4﷯ = A Hence we can write 3𝑥 −2﷮ 𝑥 + 1﷯﷮2﷯ 𝑥 + 3﷯﷯= 11﷮4 𝑥 + 1﷯﷯ − 5﷮ 2 𝑥 + 1﷯﷮2﷯﷯ − 11﷮4 𝑥 + 3﷯﷯ Therefore ﷮﷮ 3𝑥 −2﷮ 𝑥 + 1﷯﷮2﷯ 𝑥 + 3﷯﷯﷯ 𝑑𝑥= ﷮﷮ 11﷮4 𝑥 + 1﷯﷯﷯ 𝑑𝑥− ﷮﷮ 5﷮ 2 𝑥 + 1﷯﷮2﷯﷯﷯ 𝑑𝑥− ﷮﷮ 11﷮4 𝑥 + 3﷯﷯﷯ 𝑑𝑥 = 11﷮4﷯ log﷮ 𝑥+1﷯﷯− 5﷮2﷯× −1﷯﷮ 𝑥 + 1﷯﷯ − 11﷮4﷯ log﷮ 𝑥+3﷯﷯+𝐶 = 11﷮4﷯ log﷮ 𝑥+1﷯﷯− log﷮ 𝑥+3﷯﷯﷯+ 5﷮2 𝑥 + 1﷯﷯+𝐶 = 𝟏𝟏﷮𝟒﷯ 𝒍𝒐𝒈﷮ 𝒙 + 𝟏﷮𝒙 + 𝟑﷯﷯﷯ + 𝟓﷮𝟐 𝒙 + 𝟏﷯﷯+𝑪

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.