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Example 14 - Find integral x2 / (x2 + 1) (x2 + 4) dx - Integration by partial fraction - Type 1

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Example 14 Find ﷮﷮ 𝑥﷮2﷯ ﷮ 𝑥﷮2﷯ + 1﷯ 𝑥﷮2﷯ + 4﷯﷯﷯ 𝑑𝑥 Solving Integral Putting 𝑥﷮2﷯=𝑦 𝑥﷮2﷯ ﷮ 𝑥﷮2﷯ + 1﷯ 𝑥﷮2﷯ + 4﷯﷯= 𝑦﷮ 𝑦 + 1﷯ 𝑦 + 4﷯﷯ We can write this in form 𝑦﷮ 𝑦 + 1﷯ 𝑦 + 4﷯﷯= 𝐴﷮ 𝑦 + 1﷯﷯ + 𝐵﷮ 𝑦 + 4﷯﷯ 𝑦﷮ 𝑦 + 1﷯ 𝑦 + 4﷯﷯= 𝐴 𝑦 + 1﷯ + 𝐵 𝑦 + 4﷯﷮ 𝑦 + 1﷯ 𝑦 + 4﷯﷯ By cancelling denominator 𝑦=𝐴 𝑦+4﷯+𝐵 𝑦+1﷯ 𝑦=𝐴 𝑦+4﷯+𝐵 𝑦+1﷯ Hence we can write 𝑦﷮ 𝑦 + 1﷯ 𝑦 + 4﷯﷯= −1﷮3 𝑦 + 1﷯﷯ + 4﷮3 𝑦 + 4﷯﷯ Substituting back 𝑦= 𝑥﷮2﷯ 𝑥﷮2﷯ ﷮ 𝑥﷮2﷯ + 1﷯ 𝑥﷮2﷯ + 4﷯﷯= −1﷮3 𝑥﷮2﷯ + 1﷯﷯ + 4﷮3 𝑥﷮2﷯ + 4﷯﷯ Therefore, ﷮﷮ 𝑥﷮2﷯ ﷮ 𝑥﷮2﷯ + 1﷯ 𝑥﷮2﷯ + 4﷯﷯﷯ 𝑑𝑥= ﷮﷮ −1﷮3 𝑥﷮2﷯ + 1﷯﷯﷯ 𝑑𝑥+ ﷮﷮ 4﷮3 𝑥﷮2﷯ + 4﷯﷯﷯ 𝑑𝑥 = −1﷮3﷯ ﷮﷮ 1﷮3 𝑥﷮2﷯ + 1﷯﷯﷯ 𝑑𝑥+ 4﷮3﷯ ﷮﷮ 1﷮3 𝑥﷮2﷯ + 4﷯﷯﷯ 𝑑𝑥 = −1﷮3﷯ ﷮﷮ 1﷮3 𝑥﷮2﷯ + 1﷯﷯﷯ 𝑑𝑥+ 4﷮3﷯ ﷮﷮ 1﷮3 𝑥﷮2﷯ + 4﷯﷯﷯ 𝑑𝑥 = −1﷮3﷯ 𝑡𝑎𝑛﷮−1﷯﷮𝑥﷯+ 4﷮3﷯× 1﷮2﷯ 𝑡𝑎𝑛﷮−1﷯﷮ 𝑥﷮2﷯﷯ +𝐶 = −𝟏﷮𝟑﷯ 𝒕𝒂𝒏﷮−𝟏﷯﷮𝒙﷯+ 𝟐﷮𝟑﷯ 𝒕𝒂𝒏﷮−𝟏﷯﷮ 𝒙﷮𝟐﷯﷯ +𝑪

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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