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Ex 7.9, 11 - Direct Integrate dx /x2 - 1  from 2 to 3 - Definate Integration - By Partial Fraction

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex7.9, 11 2﷮3﷮ 𝑑𝑥﷮𝑥2 − 1﷯﷯ Step 1 :- Let F 𝑥﷯= ﷮﷮ 𝑑𝑥﷮ 𝑥﷮2﷯ − 1﷯﷯ We can write integrate as 1﷮ 𝑥﷮2﷯ − 1﷯= 1﷮ 𝑥 − 1﷯ 𝑥 + 1﷯﷯ 1﷮ 𝑥 − 1﷯ 𝑥 + 1﷯﷯= A﷮𝑥 − 1﷯+ B﷮ 𝑥 + 1﷯﷯ 1﷮ 𝑥 − 1﷯ 𝑥 + 1﷯﷯= A 𝑥 + 1﷯ + B 𝑥 − 1﷯﷮ 𝑥 − 1﷯ 𝑥 + 1﷯﷯ By Canceling denominator 1=A 𝑥+1﷯+B 𝑥−1﷯ Putting 𝑥=−1 1=A −1+1﷯+B −1−1﷯ 1 =A ×0+=B −2﷯ 1=−2B B= −1﷮ 2﷯ Similarly putting 𝑥=1 1=A 1+1﷯+B 1−1﷯ 1 =A 2﷯+B×0 1=2A A= 1﷮2﷯ Therefore, ﷮﷮ 1﷮ 𝑥−1﷯ 𝑥+1﷯﷯= ﷮﷮ 1 𝑑𝑥﷮2 𝑥−1﷯﷯+ ﷮﷮ −1﷮ 2﷯ 1﷮ 𝑥 + 1﷯﷯ 𝑑𝑥﷯﷯﷯ = 1﷮2﷯ ﷮﷮ 1﷮ 𝑥−1﷯﷯𝑑𝑥− ﷮﷮ 𝑑𝑥﷮ 𝑥+1﷯﷯﷯﷯ = 1﷮2﷯ 𝑙𝑜𝑔 𝑥−1﷯−𝑙𝑜𝑔 𝑥+1﷯﷯ = 1﷮2﷯𝑙𝑜𝑔 𝑥 − 1﷮𝑥 + 1﷯﷯ Hence F 𝑥﷯= 1﷮2﷯𝑙𝑜𝑔 𝑥 − 1﷮𝑥 + 1﷯﷯ Step 2 :- 2﷮3﷮ 1﷮1− 𝑥﷮2﷯﷯𝑑𝑥=𝐹 3﷯−𝐹 2﷯﷯ = 1﷮2﷯𝑙𝑜𝑔 3−1﷮3+1﷯﷯− 1﷮2﷯𝑙𝑜𝑔 2−1﷮2+1﷯﷯ = 1﷮2﷯𝑙𝑜𝑔 2﷮4﷯﷯− 1﷮2﷯𝑙𝑜𝑔 1﷮3﷯﷯ = 1﷮2﷯𝑙𝑜𝑔 1﷮2﷯﷮ 1﷮3﷯﷯﷯ = 𝟏﷮𝟐﷯𝒍𝒐𝒈 𝟑﷮𝟐﷯

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Davneet Singh
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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