Ex 7.2, 10 - Chapter 7 Class 12 Integrals

Transcript

Ex 7.2, 10 Integrate the function: 1﷮𝑥 − ﷮𝑥﷯﷯ Step 1: Simplify the given function 1﷮𝑥 − ﷮𝑥﷯﷯ = 1﷮ ﷮𝑥﷯ ﷮𝑥﷯ − 1﷯﷯ Step 2: ﷮𝑥﷯ − 1=𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 1﷮2﷯ . 𝑥﷮ 1﷮2﷯ − 1﷯−0= 𝑑𝑡﷮𝑑𝑥﷯ 1﷮2﷯ . 𝑥﷮− 1﷮2﷯﷯= 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥= 2 𝑑𝑡﷮ 𝑥﷮− 1﷮2﷯﷯﷯ 𝑑𝑥=2 ﷮𝑥﷯ 𝑑𝑡 Step 3: Integrating the function ﷮﷮ 1﷮𝑥 − ﷮𝑥﷯﷯﷯ . 𝑑𝑥 = ﷮﷮ 1﷮ ﷮𝑥﷯ ﷮𝑥﷯ − 1﷯﷯﷯ . 𝑑𝑥 putting 𝑡= ﷮𝑥﷯ − 1 & 𝑑𝑥=2 ﷮𝑥 ﷯ 𝑑𝑡 = ﷮﷮ 1﷮ ﷮𝑥﷯ 𝑡﷯﷯﷯ . 2 ﷮𝑥﷯ . 𝑑𝑥 = ﷮﷮ 2﷮𝑡﷯﷯ . 𝑑𝑡 = 2 ﷮﷮ 1﷮𝑡﷯﷯ . 𝑑𝑡 = 2 log﷮ 𝑡﷯﷯+𝐶 = 𝟐 𝒍𝒐𝒈﷮ ﷮𝒙﷯ −𝟏﷯﷯+𝑪