Ex 7.2, 7

Transcript

Ex 7.2, 7 (Method 1) Integrate the function: 𝑥 ﷮𝑥+2﷯ 𝑥 ﷮𝑥+2﷯ Step 1: Let (𝑥+2)=𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 1+0 = 𝑑𝑡﷮𝑑𝑥﷯ 1= 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥=𝑑𝑡 Step 2: Integrating the function ﷮﷮ 𝑥 ﷮𝑥+2﷯﷯ .𝑑𝑥 putting the value of 𝑥+2 & 𝑑𝑥 . = ﷮﷮𝑥 ﷮𝑡﷯﷯ .𝑑𝑥 = ﷮﷮𝑥 ﷮𝑡﷯﷯ .𝑑𝑡 = ﷮﷮ 𝑡−2﷯ ﷮𝑡﷯﷯ .𝑑𝑡 = ﷮﷮ 𝑡−2﷯ 𝑡﷮ 1﷮2﷯﷯﷯ .𝑑𝑡 = ﷮﷮ 𝑡. 𝑡﷮ 1﷮2﷯﷯−2. 𝑡﷮ 1﷮2﷯﷯﷯﷯ .𝑑𝑡 = ﷮﷮ 𝑡﷮ 3﷮2﷯﷯−2. 𝑡﷮ 1﷮2﷯﷯﷯﷯ .𝑑𝑡 = ﷮﷮ 𝑡﷮ 3﷮2﷯﷯﷯ .𝑑𝑡 − 2 ﷮﷮ 𝑡﷮ 1﷮2﷯﷯﷯ .𝑑𝑡 = 𝑡﷮ 3﷮2﷯ + 1﷯﷮ 3﷮2﷯ + 1﷯ − 2 . 𝑡﷮ 1﷮2﷯ + 1﷯﷮ 1﷮2﷯ + 1﷯ + 𝐶 = 𝑡﷮ 5﷮2﷯﷯﷮ 5﷮2﷯﷯ − 2 . 𝑡﷮ 3﷮2﷯﷯﷮ 3﷮2﷯﷯ + 𝐶 = 2﷮5﷯ . 𝑡﷮ 5﷮2﷯﷯ − 2 . 2﷮3﷯ 𝑡﷮ 3﷮2﷯﷯ + 𝐶 = 2﷮5﷯ . 𝑡﷮ 5﷮2﷯﷯ − 4﷮3﷯ 𝑡﷮ 3﷮2﷯﷯ + 𝐶 = 𝟐﷮𝟓﷯ . 𝒙+𝟐﷯﷮ 𝟓﷮𝟐﷯﷯ − 𝟒﷮𝟑﷯ 𝒙+𝟐﷯﷮ 𝟑﷮𝟐﷯﷯ + 𝑪 Ex 7.2, 7 (Method 2) Integrate the function: 𝑥 ﷮𝑥+2﷯ 𝑥 ﷮𝑥+2﷯ = 𝑥+2−2﷯ ﷮𝑥+2﷯ = 𝑥+2﷯−2﷯ ﷮𝑥+2﷯ = 𝑥+2﷯ ﷮𝑥+2﷯−2 ﷮𝑥+2﷯ = 𝑥+2﷯ 𝑥+2﷯﷮ 1﷮2﷯﷯−2 𝑥+2﷯﷮ 1﷮2﷯﷯ = 𝑥+2﷯﷮ 1﷮2﷯ +1﷯−2 𝑥+2﷯﷮ 1﷮2﷯﷯ = 𝑥+2﷯﷮ 3﷮2﷯﷯−2 𝑥+2﷯﷮ 1﷮2﷯﷯ ∴ 𝑥 ﷮𝑥+2﷯ = 𝑥+2﷯﷮ 3﷮2﷯﷯−2 𝑥+2﷯﷮ 1﷮2﷯﷯ Step 2: Integrating the function ﷮﷮ 𝑥 ﷮𝑥+2﷯﷯ = ﷮﷮ 𝑥+2﷯﷮ 3﷮2﷯﷯−2 𝑥+2﷯﷮ 1﷮2﷯﷯﷯﷯. 𝑑𝑥 = ﷮﷮ 𝑥+2﷯﷮ 3﷮2﷯﷯﷯. 𝑑𝑥 − 2 ﷮﷮ 𝑥+2﷯﷮ 1﷮2﷯﷯﷯. 𝑑𝑥 = 𝑥 + 2﷯﷮ 3﷮2﷯ + 1﷯﷮ 3﷮2﷯ + 1﷯ − 2 𝑥+2﷯﷮ 1﷮2﷯ + 1﷯ ﷮ 1﷮2﷯ + 1﷯ + 𝐶 = 𝑥 + 2﷯﷮ 5﷮2﷯﷯﷮ 5﷮2﷯﷯ − 2 𝑥 + 2﷯﷮ 3﷮2﷯﷯ ﷮ 3﷮2﷯﷯ + 𝐶 = 2﷮5﷯ 𝑥 + 2﷯﷮ 5﷮2﷯﷯ − 2 . 2 ﷮3﷯ 𝑥 + 2﷯﷮ 3﷮2﷯﷯ + 𝐶 = 𝟐﷮𝟓﷯ 𝒙 + 𝟐﷯﷮ 𝟓﷮𝟐﷯﷯ − 𝟒﷮𝟑﷯ 𝒙 + 𝟐﷯﷮ 𝟑﷮𝟐﷯﷯ + 𝑪