Ex 7.2, 34 - Integrate root(tan x) / sin x cos x

Ex 7.2, 34 ﷮ tan﷮𝑥﷯﷯﷮ sin﷮𝑥 cos﷮𝑥﷯﷯﷯ ﷮ tan﷮𝑥﷯﷯﷮ sin﷮𝑥 cos﷮𝑥﷯﷯﷯ The given function cannot be integrated by direct substitution, Step 1: Simplify the given function ﷮ tan﷮𝑥﷯﷯﷮ sin﷮𝑥 cos﷮𝑥﷯﷯﷯ = ﷮ tan﷮𝑥﷯﷯﷮ sin﷮𝑥 cos﷮𝑥﷯﷯. cos﷮𝑥﷯﷮ cos﷮𝑥﷯﷯﷯ = ﷮ tan﷮𝑥﷯﷯﷮ sin﷮𝑥﷯ . cos﷮2﷯﷮𝑥﷯﷮ cos﷮𝑥﷯﷯﷯ = ﷮ tan﷮𝑥﷯﷯﷮ cos﷮2﷯﷮𝑥﷯ . sin 𝑥﷮ cos﷮𝑥﷯﷯﷯ = ﷮ tan﷮𝑥﷯﷯﷮ cos﷮2﷯﷮𝑥﷯ . tan﷮𝑥﷯﷯ = tan﷮𝑥﷯﷯﷮ 1﷮2﷯ − 1﷯ × 1﷮ cos﷮2﷯﷮𝑥﷯﷯ = tan﷮𝑥﷯﷯﷮ −1﷮2﷯﷯ × 1﷮ cos﷮2﷯﷮𝑥﷯﷯ = tan﷮𝑥﷯﷯﷮ −1﷮2﷯﷯ × sec﷮2﷯﷮𝑥﷯ ∴ ﷮ tan﷮𝑥﷯﷯﷮ sin﷮𝑥 cos﷮𝑥﷯﷯﷯ = tan﷮𝑥﷯﷯﷮ −1﷮2﷯﷯ × sec﷮2﷯﷮𝑥﷯ Step 2: Integrating the function ﷮﷮ ﷮ tan﷮𝑥﷯﷯﷮ sin﷮𝑥 cos﷮𝑥﷯﷯﷯﷯ . 𝑑𝑥 = ﷮﷮ tan﷮𝑥﷯﷯﷮ −1﷮2﷯﷯ × sec﷮2﷯﷮𝑥﷯﷯. 𝑑𝑥 Let tan⁡𝑥 = 𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 sec﷮2﷯﷮𝑥﷯= 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥= 𝑑𝑡﷮ sec﷮2﷯﷮𝑥﷯﷯ Thus, our equation becomes ∴ ﷮﷮ tan﷮𝑥﷯﷯﷮ −1﷮2﷯﷯ . sec﷮2﷯﷮𝑥﷯﷯. 𝑑𝑥 = ﷮﷮ 𝑡﷯﷮ −1﷮2﷯﷯ . sec﷮2﷯﷮𝑥﷯﷯. 𝑑𝑡﷮ sec﷮2﷯﷮𝑥﷯﷯ = ﷮﷮ 𝑡﷮ −1﷮2﷯﷯ . 𝑑𝑡﷯ = 𝑡﷮− 1﷮2﷯ +1﷯﷮− 1﷮2﷯ +1﷯ + 𝐶 = 𝑡﷮ 1﷮2﷯﷯﷮ 1﷮2﷯﷯ + 𝐶 = 2𝑡﷮ 1﷮2﷯﷯+ 𝐶 = 2 ﷮𝑡﷯+ 𝐶 = 𝟐 ﷮ 𝐭𝐚𝐧﷮𝒙﷯﷯+ 𝑪