Misc 3 - Integrate 1 / x root ax + x2

Misc 3 Integrate the function 1﷮𝑥 ﷮𝑎𝑥 + 𝑥﷮2﷯﷯ ﷯ ﷮﷮ 1﷮𝑥 ﷮𝑎𝑥 + 𝑥﷮2﷯﷯ ﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮𝑥 ﷮ 𝑥﷮2﷯ 𝑎﷮𝑥﷯ − 1﷯﷯ ﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮𝑥 . 𝑥 ﷮ 𝑎﷮𝑥﷯ − 1﷯﷯ ﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ 𝑥﷮2﷯ ﷮ 𝑎﷮𝑥﷯ − 1﷯﷯ ﷯﷯ 𝑑𝑥 Let 𝑎﷮𝑥﷯ −1=𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 − 𝑎﷮ 𝑥﷮2﷯﷯ −0 = 𝑑𝑡﷮𝑑𝑥﷯ − 𝑎﷮ 𝑥﷮2﷯﷯ = 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥 = − 𝑥﷮2﷯﷮𝑎﷯ . 𝑑𝑡 Putting the values of a﷮x﷯−1﷯ and dx, we get ﷮﷮ 1﷮ 𝑥﷮2﷯ ﷮ 𝑎﷮𝑥﷯ − 1﷯﷯ ﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ 𝑥﷮2﷯ ﷮𝑡﷯ ﷯﷯ 𝑑𝑥 = ﷮﷮ 1﷮ 𝑥﷮2﷯ ﷮𝑡﷯ ﷯﷯× − 𝑥﷮2﷯﷮𝑎﷯ . 𝑑𝑡 = ﷮﷮ −1﷮𝑎﷯﷯ 𝑑𝑡﷮ ﷮𝑡﷯﷯ = −1﷮𝑎﷯ ﷮﷮ 𝑡﷯﷮− 1﷮2﷯﷯﷯ 𝑑𝑡 = −1﷮𝑎﷯ 𝑡﷮− 1﷮2﷯ + 1﷯﷮− 1﷮2﷯ + 1﷯﷯+𝐶 = −1﷮𝑎﷯ 𝑡﷮ 1﷮2﷯﷯﷮ 1﷮2﷯﷯﷯ + 𝐶 = −2﷮𝑎﷯ ﷮𝑡﷯﷯ + 𝐶 Putting back t = 𝑎﷮𝑥﷯−1 = −2﷮𝑎﷯ ﷮ 𝑎﷮𝑥﷯ −1﷯ + 𝐶 = −𝟐﷮𝒂﷯ ﷮ 𝒂 − 𝒙﷮𝒙﷯﷯ + 𝑪

Misc 3 - Chapter 7 Class 12 Integrals