Misc 5 - Integrate 1 / x1/2 + x1/3 - Class 12 NCERT

Misc 5 Integrate the function 1﷮ 𝑥﷮ 1﷮2﷯﷯ + 𝑥﷮ 1﷮3﷯﷯ ﷯ ﷮﷮ 1﷮ 𝑥﷮ 1﷮2﷯﷯ + 𝑥﷮ 1﷮3﷯﷯﷯﷯ 𝑑𝑥 Let x = 𝑡﷮6﷯ 𝑑𝑥﷮𝑑𝑡﷯=6 𝑡﷮5﷯ dx = 6 𝑡﷮5﷯ dt Substituting value of x and dx = ﷮﷮ 6𝑡﷮5﷯﷮ (𝑡﷮6﷯) ﷮ 1﷮2﷯﷯ + (𝑡﷮6﷯) ﷮ 1﷮3﷯﷯﷯﷯ = ﷮﷮ 6𝑡﷮5﷯﷮ 𝑡﷮3﷯ + 𝑡﷮2﷯﷯﷯ 𝑑𝑡 = 6 ﷮﷮ 𝑡﷮3﷯﷮𝑡 + 1﷯﷯ 𝑑𝑡 Adding and subtracting 1 in numerator = 6 ﷮﷮ 𝑡﷮3﷯ + 1 − 1﷮𝑡 + 1﷯﷯ 𝑑𝑡 = 6 ﷮﷮ 𝑡﷮3﷯ + 1 ﷮𝑡 + 1﷯− 1﷮𝑡 + 1﷯﷯﷯ 𝑑𝑡 = 6 ﷮﷮ 𝑡 + 1﷯( 𝑡﷮2﷯+ 1 − 𝑡)﷮𝑡 + 1﷯﷯− 1﷮𝑡 + 1﷯ 𝑑𝑡 = ﷮﷮ 𝑡﷮2﷯+1−𝑡﷯− 1﷮1 + 𝑡﷯﷯ 𝑑𝑡 = 6 𝑡﷮3﷯﷮3﷯+𝑡− 𝑡﷮2﷯﷮2﷯−𝑙𝑜𝑔 1+𝑡﷯﷯+ C = 2 𝑡﷮3﷯+6𝑡− 3𝑡﷮2﷯−6 𝑙𝑜𝑔 1+𝑡﷯+ C Putting back value of t = 2 (𝑥﷮ 1﷮6﷯﷯)﷮3﷯+6 (𝑥﷮ 1﷮6﷯﷯)−3 (𝑥﷮ 1﷮6﷯﷯)﷮2﷯−6 𝑙𝑜𝑔 1+ 𝑥﷮ 1﷮6﷯﷯﷯+ C = 2 𝑥﷮ 1﷮2﷯﷯+6 𝑥﷮ 1﷮6﷯﷯−3 𝑥﷮ 1﷮3﷯﷯−6 𝑙𝑜𝑔 1+ 𝑥﷮ 1﷮6﷯﷯﷯+ C = 2 ﷮𝑥﷯−3 𝑥﷮ 1﷮3﷯﷯+ 6 𝑥﷮ 1﷮6﷯﷯−6 log﷮(1+ 𝑥 ﷮ 1﷮6﷯﷯)﷯+ C