Ex 7.10, 6 - Evaluate integrals dx / x + 4 - x2 - Definate Integration - By Substitution

Slide16.JPG
Slide17.JPG Slide18.JPG Slide19.JPG

  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
Ask Download

Transcript

Ex7.10, 6 Evaluate the integrals using substitution 0﷮2 ﷮ 𝑑𝑥﷮𝑥 + 4 − 𝑥﷮2﷯﷯﷯ We can write 0﷮2﷮ 𝑑𝑥﷮𝑥 + 4 − 𝑥﷮2﷯﷯= 0﷮2﷮ 𝑑𝑥﷮− 𝑥﷮2﷯ − 𝑥 − 4﷯﷯﷯﷯ =− 0﷮2﷮ 𝑑𝑥﷮ 𝑥﷮2﷯ − 𝑥 − 4﷯﷯ =− 0﷮2﷮ 𝑑𝑥﷮ 𝑥﷮2﷯ −2 × 1﷮2﷯ × 𝑥 − 4﷯﷯ =− 0﷮2﷮ 𝑑𝑥﷮ 𝑥﷮2﷯ −2 × 1﷮2﷯ × 𝑥 + 1﷮ 2﷮2﷯﷯ − 1﷮ 2﷮2﷯﷯ − 4﷯﷯ =− 0﷮2﷮ 𝑑𝑥﷮ 𝑥 − 1﷮2﷯﷯﷮2﷯− 1﷮4﷯ − 4﷯﷯ =− 0﷮2﷮ 𝑑𝑥﷮ 𝑥 − 1﷮2﷯﷯﷮2﷯− 17﷮4﷯ ﷯﷯ =− 0﷮2﷮ 𝑑𝑥﷮ 𝑥 − 1﷮2﷯﷯﷮2﷯− ﷮17﷯﷮4﷯﷯﷮2﷯ ﷯﷯ Let 𝑡=𝑥− 1﷮2﷯ Differentiating w.r.t.𝑥 𝑑𝑡﷮𝑑𝑥﷯=1 𝑑𝑡=𝑑𝑥 When x varies from 0 to 2, then t varies from −1﷮2﷯ to 3﷮2﷯. Therefore, − 0﷮2﷮ 𝑑𝑥﷮ 𝑥 − 1﷮2﷯﷯﷮2﷯− ﷮17﷯﷮2﷯﷯﷮2﷯﷯=− −1﷮2﷯﷮ 3﷮2﷯﷮ 𝑑𝑡﷮𝑡 − ﷮17﷯﷮2﷯﷯﷮2﷯﷯﷯﷯ =− 1﷮2 ﷮17﷯﷮2﷯﷯﷯𝑙𝑜𝑔 𝑡 − ﷮17﷯﷮2﷯﷮𝑡 + ﷮17﷯﷮2﷯﷯﷯﷯﷮ −1﷮ 2﷯﷮ 3﷮2﷯﷯ =− 1﷮ ﷮17﷯﷯ 𝑙𝑜𝑔 3﷮2﷯ − ﷮17﷯﷮2﷯﷮ 3﷮2﷯ + ﷮17﷯﷮2﷯﷯﷯+𝑙𝑜𝑔 −1﷮ 2﷯ − ﷮17﷯﷮2﷯﷮ −1﷮ 2﷯ + ﷮17﷯﷮2﷯﷯﷯﷯ =− 1﷮ ﷮17﷯﷯ 𝑙𝑜𝑔 3 − ﷮17﷯﷮3 + ﷮17﷯﷯﷯+𝑙𝑜𝑔 − 1 + ﷮17﷯﷯﷮− 1 − ﷮17﷯﷯﷯﷯﷯ =− 1﷮ ﷮17﷯﷯𝑙𝑜𝑔 3 − ﷮17﷯﷮3 + ﷮17﷯﷯﷮ 1 + ﷮17﷯﷮1 − ﷮17﷯﷯﷯﷯ =− 1﷮ ﷮17﷯﷯𝑙𝑜𝑔 3 − ﷮17﷯﷮3 + ﷮17﷯﷯ × 1 − ﷮17﷯﷮1 + ﷮17﷯﷯﷯ =− 1﷮ ﷮17﷯﷯𝑙𝑜𝑔 3+17 − 3 ﷮17﷯ − ﷮17﷯﷮3 +17 + 3 ﷮17﷯ + ﷮17﷯﷯ ﷯ =− 1﷮ ﷮17﷯﷯𝑙𝑜𝑔 20 − 4 ﷮17﷯﷮20 + 4 ﷮17﷯﷯ ﷯ =− 1﷮ ﷮17﷯﷯𝑙𝑜𝑔 4 5 − ﷮17﷯﷯﷮4 5 + ﷮17﷯﷯﷯ ﷯ =− 1﷮ ﷮17﷯﷯𝑙𝑜𝑔 5 − ﷮17﷯﷮5 + ﷮17﷯﷯ ﷯ = 1﷮ ﷮17﷯﷯𝑙𝑜𝑔 5 − ﷮17﷯﷮5 + ﷮17﷯﷯ ﷯﷮−1﷯ = 1﷮ ﷮17﷯﷯𝑙𝑜𝑔 5 + ﷮17﷯﷮5 − ﷮17﷯﷯﷯ = 1﷮ ﷮17﷯﷯𝑙𝑜𝑔 5 + ﷮17﷯﷮5 − ﷮17﷯﷯ × 5 + ﷮17﷯﷮5 + ﷮17﷯﷯﷯ = 1﷮ ﷮17﷯﷯𝑙𝑜𝑔 5 − ﷮17﷯﷯﷮2﷯﷮ 5﷮2﷯ − ﷮17﷯﷯﷮2﷯﷯ ﷯ = 1﷮ ﷮17﷯﷯𝑙𝑜𝑔 25 + 17 + 10 ﷮17﷯﷮25 − 17﷯ ﷯ = 1﷮ ﷮17﷯﷯𝑙𝑜𝑔 42 + 10 ﷮17﷯﷮8﷯ ﷯ = 𝟏﷮ ﷮𝟏𝟕﷯﷯𝒍𝒐𝒈 𝟐𝟏 + 𝟓 ﷮𝟏𝟕﷯﷮𝟒﷯ ﷯

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.
Jail