Ex 7.9, 13 - Direct Integrate x dx / x2 + 1 from 2 to 3 - Ex 7.9

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex7.9, 13 2﷮3﷮ 𝑥𝑑𝑥﷮𝑥2+1﷯﷯ Step 1 :- Let F 𝑥﷯= ﷮﷮ 𝑥﷮ 𝑥﷮2﷯ + 1﷯ 𝑑𝑥﷯ Let 𝑥﷮2﷯ + 1=𝑡 Differentiating w.r.t.𝑥 𝑑﷮𝑑𝑥﷯ 𝑥﷮2﷯+1﷯= 𝑑𝑡﷮𝑑𝑥﷯ 2𝑥= 𝑑𝑡﷮𝑑𝑥﷯ d𝑥= 𝑑𝑡﷮2𝑥﷯ Putting Values of 𝑥﷮2﷯+1=𝑡 & 𝑑𝑥= 𝑑𝑡﷮2𝑥﷯ ﷮﷮ 𝑥 𝑑𝑥﷮ 𝑥﷮2﷯ + 1﷯= ﷮﷮ 𝑥﷮𝑡﷯ 𝑑𝑡﷮2𝑥﷯﷯﷯ = 1﷮2﷯ ﷮﷮ 𝑑𝑡﷮𝑡﷯﷯ = 1﷮2﷯𝑙𝑜𝑔 𝑡﷯ = 1﷮2﷯𝑙𝑜𝑔 𝑥﷮2﷯+1﷯ Hence F 𝑥﷯= 1﷮2﷯𝑙𝑜𝑔 𝑥﷮2﷯+1﷯ Step 2 :- 2﷮3﷮ 𝑥 𝑑𝑥﷮ 𝑥﷮2﷯+1﷯﷯=𝐹 3﷯−𝐹 2﷯ = 1﷮2﷯𝑙𝑜𝑔 3﷮2﷯+1﷯− 1﷮2﷯𝑙𝑜𝑔 2﷮2﷯+1﷯ = 1﷮2﷯ log﷮10− 1﷮2﷯𝑙𝑜𝑔 5﷯﷯ = 1﷮2﷯ log﷮10− log﷮5﷯﷯﷯ = 1﷮2﷯ 𝑙𝑜𝑔 10﷮5﷯ = 𝟏﷮𝟐﷯ 𝐥𝐨𝐠﷮𝟐﷯

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