Ex 7.2, 19 - Integrate e2x - 1 / e2x + 1 - Chapter 7

Ex 7.2, 19 𝑒2𝑥 −1﷮𝑒2𝑥+1﷯ Step 1: Simplify the given function 𝑒﷮2𝑥﷯ − 1﷮ 𝑒﷮2𝑥﷯ + 1﷯ Dividing numerator and denominator by ex, we obtain = 𝑒﷮2𝑥﷯﷮ 𝑒﷮𝑥﷯﷯ − 𝟏﷮ 𝒆﷮𝒙﷯﷯﷮ 𝑒﷮2𝑥﷯﷮ 𝑒﷮𝑥﷯﷯ + 𝟏﷮ 𝒆﷮𝒙﷯﷯2﷯ = 𝑒﷮𝟐𝒙 −𝒙﷯ − 1 . 𝒆﷮−𝒙﷯﷮ 𝑒﷮𝟐𝒙 −𝒙﷯ + 1 . 𝒆﷮−𝒙﷯﷯ = 𝑒﷮𝒙﷯ − 𝒆﷮−𝒙﷯﷮ 𝑒﷮𝒙﷯ + 𝒆﷮−𝒙﷯﷯ ∴ 𝑒﷮2𝑥﷯ − 1﷮ 𝑒﷮2𝑥﷯ + 1﷯ = 𝑒﷮𝑥﷯ − 𝑒﷮−𝑥﷯﷮ 𝑒﷮𝑥﷯ + 𝑒﷮−𝑥﷯﷯ Step 2: Let 𝑒﷮𝑥﷯ + 𝑒﷮−𝑥﷯= 𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑒﷮𝑥﷯+ −1﷯ 𝑒﷮−𝑥﷯= 𝑑𝑡﷮𝑑𝑥﷯ 𝑒﷮𝑥﷯− 𝑒﷮−𝑥﷯= 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥= 𝑑𝑡﷮ 𝑒﷮𝑥﷯ − 𝑒﷮−𝑥﷯﷯ Step 3: Integrating the function ﷮﷮ 𝑒﷮2𝑥﷯ − 1﷮ 𝑒﷮2𝑥﷯ + 1﷯ ﷯. 𝑑𝑥 = ﷮﷮ 𝑒﷮𝑥﷯ − 𝑒﷮−𝑥﷯﷮ 𝑒﷮𝑥﷯ + 𝑒﷮−𝑥﷯﷯ ﷯. 𝑑𝑥 Putting 𝑒﷮𝑥﷯ + 𝑒﷮−𝑥﷯=𝑡 & 𝑑𝑥= 𝑑𝑡﷮ 𝑒﷮𝑥﷯ − 𝑒﷮−𝑥﷯﷯ = ﷮﷮ 𝑒﷮𝑥﷯ − 𝑒﷮−𝑥﷯﷮𝑡﷯ ﷯. 𝑑𝑡﷮ 𝑒﷮𝑥﷯ − 𝑒﷮−𝑥﷯﷯ = ﷮﷮ 1﷮𝑡﷯ ﷯. 𝑑𝑡 = log﷮ 𝑡﷯﷯+𝐶 = log﷮ 𝑒﷮𝑥﷯+ 𝑒﷮−𝑥﷯﷯﷯+𝐶 = 𝒍𝒐𝒈﷮ 𝒆﷮𝒙﷯ + 𝒆﷮−𝒙﷯﷯﷯+𝑪

Ex 7.2, 19 - Class 12