Misc 1 - integrate 1 / x - x3 - Chapter 7 Integrals

Misc 1 Integrate the function 1﷮𝑥 − 𝑥﷮3﷯﷯ We can write as : 1﷮𝑥 − 𝑥﷮3﷯﷯= 1﷮ 𝑥﷮3﷯﷯﷮ 𝑥 − 𝑥﷮3﷯﷯﷮ 𝑥﷮3﷯﷯﷯ = 1﷮ 𝑥﷮3﷯ 𝑥﷮ 𝑥﷮3﷯﷯ − 𝑥﷮3﷯﷮ 𝑥﷮3﷯﷯﷯﷯ = 1﷮ 𝑥﷮3﷯ 1﷮ 𝑥﷮2﷯﷯ −1﷯﷯ Integrating w.r.t.x ﷮﷮ 1﷮𝑥 − 𝑥﷮3﷯﷯𝑑𝑥﷯= ﷮﷮ 1﷮ 𝑥﷮3﷯ 1﷮ 𝑥﷮2﷯﷯ − 1﷯﷯﷯ 𝑑𝑥 Let t = 1﷮ 𝑥﷮2﷯﷯−1 Differentiating with respect to 𝑥 𝑑﷮𝑑𝑥﷯ 1﷮ 𝑥﷮2﷯﷯−1﷯= 𝑑𝑡﷮𝑑𝑥﷯ −2﷮ 𝑥﷮3﷯﷯= 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥= 𝑥﷮3﷯𝑑𝑡﷮−2﷯ Substituting value of t ﷮﷮ 1﷮ 𝑥﷮3﷯ 1﷮ 𝑥﷮2﷯﷯ −1﷯﷯𝑑𝑥﷯= ﷮﷮ 1﷮ 𝑥﷮3﷯ 𝑡﷯﷯ × 𝑥﷮3﷯𝑑𝑡﷮−2﷯﷯ = ﷮﷮ 1﷮−2﷯ 𝑑𝑡﷮𝑡﷯﷯ = −1﷮ 2﷯ ﷮﷮ 𝑑𝑡﷮𝑡﷯﷯ = −1﷮ 2﷯𝑙𝑜𝑔 𝑡﷯+𝐶 Putting back 𝑡= 1﷮ 𝑥﷮2﷯﷯−1 = −1﷮ 2﷯𝑙𝑜𝑔 1﷮ 𝑥﷮2﷯﷯−1﷯+𝐶 = −1﷮ 2﷯𝑙𝑜𝑔 1﷮ 𝑥﷮2﷯﷯− 𝑥﷮2﷯﷮ 𝑥﷮2﷯﷯﷯+𝐶 = −1﷮ 2﷯𝑙𝑜𝑔 1 − 𝑥﷮2﷯﷮ 𝑥﷮2﷯﷯﷯+𝐶 = 1﷮2﷯ log 1 − 𝑥﷮2﷯﷮ 𝑥﷮2﷯﷯﷯﷮−1﷯+𝐶 = 𝟏﷮𝟐﷯ log 𝒙﷮𝟐﷯﷮𝟏 − 𝒙﷮𝟐﷯﷯﷯+𝑪

Misc 1 - Class 12