Ex 7.4, 15 - Chapter 7 Class 12 Integrals

Transcript

Ex 7.4, 15 1﷮ ﷮(𝑥 − 𝑎)(𝑥 − 𝑏)﷯﷯ Integrating the function 𝑤.𝑟.𝑡.𝑥 ﷮﷮ 1﷮ ﷮ 𝑥 − 𝑎﷯ (𝑥 − 𝑏)﷯﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ ﷮𝑥 𝑥 − 𝑎﷯ − 𝑎(𝑥 − 𝑏)﷯﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ ﷮ 𝑥﷮2﷯ − 𝑏𝑥 − 𝑎𝑥 + 𝑎𝑏﷯﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ ﷮ 𝑥﷮2﷯ − 𝑥 𝑎 + 𝑏﷯ + 𝑎𝑏﷯﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ ﷮ 𝑥﷮2﷯ − 2 𝑥﷯ 𝑎 + 𝑏﷮2﷯﷯ + 𝑎𝑏﷯﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ ﷮ 𝑥﷮2﷯ − 2 𝑥﷯ 𝑎 + 𝑏﷮2﷯﷯ + 𝑎 + 𝑏﷮2﷯﷯﷮2﷯− 𝑎 + 𝑏﷮2﷯﷯﷮2﷯+ 𝑎𝑏﷯﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ ﷮ 𝑥 − 𝑎 + 𝑏﷮2﷯﷯﷮2﷯ − 𝑎 + 𝑏﷮2﷯﷯﷮2﷯+ 𝑎𝑏﷯﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ ﷮ 𝑥 − 𝑎 + 𝑏﷮2﷯﷯﷮2﷯ − 𝑎﷮2﷯ + 𝑏﷮2﷯+ 2𝑎𝑏﷮4﷯﷯ + 𝑎𝑏﷯﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ ﷮ 𝑥 − 𝑎 + 𝑏﷮2﷯﷯﷮2﷯+ − 𝑎﷮2﷯ − 𝑏﷮2﷯ − 2𝑎𝑏 + 4𝑎𝑏﷮4﷯﷯﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ ﷮ 𝑥 − 𝑎 + 𝑏﷮2﷯﷯﷮2﷯ + − 𝑎﷮2﷯ − 𝑏﷮2﷯ + 2𝑎𝑏﷮4﷯﷯﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ ﷮ 𝑥 − 𝑎 + 𝑏﷮2﷯﷯﷮2﷯ − 𝑎﷮2﷯ + 𝑏﷮2﷯ − 2𝑎𝑏﷮4﷯﷯﷯ ﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ ﷮ 𝑥 − 𝑎 + 𝑏﷮2﷯﷯﷮2﷯ − 𝑎 − 𝑏﷮2﷯﷯﷮2﷯﷯ ﷯﷯ 𝑑𝑥 = 𝑙𝑜𝑔﷮ 𝑥 − 𝑎 + 𝑏﷮2﷯ + ﷮ 𝑥 − 𝑎 + 𝑏﷮2﷯﷯﷮2﷯− 𝑎 − 𝑏﷮2﷯﷯﷮2﷯﷯﷯﷯+𝐶 = 𝑙𝑜𝑔﷮ 𝑥− 𝑎 + 𝑏﷮2﷯ + ﷮ 𝑥﷮2﷯+ 𝑎 + 𝑏﷮2﷯﷯﷮2﷯−2 𝑥﷯ 𝑎 + 𝑏﷮2﷯﷯− 𝑎 − 𝑏﷮2﷯﷯﷮2﷯﷯﷯﷯+𝐶 = 𝑙𝑜𝑔﷮ 𝑥− 𝑎 + 𝑏﷮2﷯ + ﷮ 𝑥﷮2﷯−2 𝑥﷯ 𝑎 + 𝑏﷮2﷯﷯+ 𝑎 + 𝑏﷮2﷯﷯﷮2﷯− 𝑎 − 𝑏﷮2﷯﷯﷮2﷯﷯﷯﷯+𝐶 = 𝑙𝑜𝑔﷮ 𝑥− 𝑎 + 𝑏﷮2﷯ + ﷮ 𝑥﷮2﷯−𝑥 𝑎+𝑏﷯+ 𝑎﷮2﷯ + 𝑏﷮2﷯ + 2𝑎𝑏﷮4﷯− 𝑎﷮2﷯ + 𝑏﷮2﷯ − 2𝑎𝑏﷮4﷯﷯﷯﷯+𝐶 = 𝑙𝑜𝑔﷮ 𝑥− 𝑎 + 𝑏﷮2﷯ + ﷮ 𝑥﷮2﷯−𝑥 𝑎+𝑏﷯+ 4𝑎𝑏﷮4﷯﷯ ﷯﷯+𝐶 = 𝑙𝑜𝑔﷮ 𝑥− 𝑎 + 𝑏﷮2﷯ + ﷮ 𝑥﷮2﷯−𝑥 𝑎+𝑏﷯+𝑎𝑏﷯ ﷯﷯+𝐶 = 𝑙𝑜𝑔﷮ 𝑥− 𝑎 + 𝑏﷮2﷯ + ﷮ 𝑥﷮2﷯−𝑎𝑥−𝑏𝑥+𝑎𝑏﷯ ﷯﷯+𝐶 = 𝑙𝑜𝑔﷮ 𝑥− 𝑎 + 𝑏﷮2﷯ + ﷮𝑥 𝑥−𝑎﷯−𝑏 𝑥−𝑎﷯﷯ ﷯﷯+𝐶 = 𝒍𝒐𝒈﷮ 𝒙− 𝒂 + 𝒃﷮𝟐﷯ + ﷮ 𝒙−𝒂﷯ 𝒙−𝒃﷯﷯ ﷯﷯+𝑪