Example 16 - Find integral x2 + x + 1 dx / (x + 2) (x2 + 1) - Examples

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Example 16 Find ﷮﷮ 𝑥﷮2﷯+ 𝑥 +1 𝑑𝑥 ﷮ 𝑥 + 2﷯ 𝑥﷮2﷯+1﷯﷯﷯ By partial fraction 𝑥﷮2﷯+ 𝑥 + 1﷮ 𝑥 + 1﷯ 𝑥 + 2﷯﷯= 𝐴﷮𝑥 + 2﷯ + 𝐵𝑥 + 𝐶﷮ 𝑥﷮2﷯+ 1﷯ Cancelling denominator 𝑥﷮2﷯+ 𝑥+1=𝐴 𝑥﷮2﷯+1﷯+ 𝐵𝑥+𝐶﷯ 𝑥+2﷯ Putting x = −𝟐 −2﷯﷮2﷯+ −2﷯+1=𝐴 −2﷯﷮2﷯+1﷯+0 4−2+1= 5A 3﷮5﷯ = A Putting x = 𝟎 0+0+0= A(0 + 1) + (0 + C) (0 + 2) 1 = A + 2C 1 = 3﷮5﷯ + 2C 1 – 3﷮5﷯ = 2C 2﷮5﷯ = 2C C = 1﷮5﷯ Thus, 𝑥﷮2﷯+ 𝑥 + 1﷮ 𝑥 + 1﷯( 𝑥﷮2﷯+ 1)﷯ = 3﷮5 (𝑥 + 2)﷯ + 1 (2𝑥 + 1)﷮5 ( 𝑥﷮2﷯ + 1)﷯ Hence, our equation becomes ﷮﷮ 𝑥﷮2﷯+ 𝑥 + 1﷮ 𝑥 + 2﷯ ( 𝑥﷮2﷯ + 1)﷯﷯ 𝑑𝑥= ﷮﷮ 3﷮5( 𝑥﷮2﷯ + 1)﷯﷯ 𝑑𝑥+ ﷮﷮ 1﷮5﷯﷯ (2𝑥 + 1)﷮ 𝑥﷮2﷯ + 1﷯ 𝑑𝑥 = ﷮﷮ 3﷮5( 𝑥﷮2﷯ + 1)﷯﷯ 𝑑𝑥+ 1﷮5﷯ ﷮﷮ 2𝑥﷮ 𝑥﷮2﷯ + 1﷯ 𝑑𝑥+﷯ 1﷮5﷯ ﷮﷮ 1﷮ 𝑥﷮2﷯ + 1﷯﷯ 𝑑𝑥 Hence ﷮﷮ 𝑥﷮2﷯+ 𝑥 + 1﷮ 𝑥 + 2﷯ ( 𝑥﷮2﷯+ 1)﷯﷯ 𝑑𝑥 = 3﷮5﷯𝑙𝑜𝑔 𝑥+2﷯+ 1﷮5﷯𝑙𝑜𝑔 𝑥﷮2﷯+1﷯+ 1﷮5﷯ 𝑡𝑎𝑛﷮−1﷯ 𝑥﷯+ C Where C = C﷮1﷯+ C﷮2﷯+ C﷮3﷯

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