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Ex 7.10, 9 - Value of integral (x - x3)1/3 / x4 - Ex 7.10

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex7.10, 9 Choose the correct answer The value of the integral 1﷮3﷯﷮1﷮ 𝑥 − 𝑥﷮3﷯﷯﷮ 1﷮3﷯﷯﷮ 𝑥﷮4﷯﷯ ﷯ 𝑑𝑥 is • 6 (B) 0 (C) 3 (D) 4 1﷮3﷯﷮1﷮ 𝑥 − 𝑥﷮3﷯﷯﷮ 1﷮3﷯﷯﷮ 𝑥﷮4﷯﷯ ﷯ 𝑑𝑥 Taking common 𝑥﷮3﷯ from numerator = 1﷮3﷯﷮1﷮ 𝑥﷮3﷯﷯﷮ 1﷮3﷯﷯ 1﷮ 𝑥﷮2﷯﷯ −1﷯﷮ 1﷮3﷯﷯﷮ 𝑥﷮4﷯﷯ ﷯ 𝑑𝑥 = 1﷮3﷯﷮1﷮ 𝑥 1﷮ 𝑥﷮2﷯﷯ −1﷯﷮ 1﷮3﷯﷯﷮ 𝑥﷮4﷯﷯ ﷯ 𝑑𝑥 = 1﷮3﷯﷮1﷮ 1﷮ 𝑥﷮2﷯﷯ −1﷯﷮ 1﷮3﷯﷯﷮ 𝑥﷮3﷯﷯ ﷯ 𝑑𝑥 Let t = 1﷮ 𝑥﷮2﷯﷯ −1 𝑑𝑡﷮𝑑𝑥﷯= −2﷮ 𝑥﷮3﷯﷯ −𝑑𝑡﷮2﷯= 𝑑𝑥﷮ 𝑥﷮3﷯﷯ Thus, when x varies from 1﷮3﷯ to 1, t varies form 0 to 8 Substituting values, 1﷮2﷯ 8﷮0﷮ 𝑡﷮ 1﷮3﷯﷯𝑑𝑡﷯ = −1﷮2﷯ 𝑡﷮ 1﷮3﷯ + 1﷯﷮ 1﷮3﷯+1﷯﷯﷮8﷮0﷯ = −1﷮2﷯ 3𝑡﷮ 4﷮3﷯ ﷯﷮4﷯﷯﷮8﷮0﷯ Putting limits, = −1﷮2﷯ 0− 3 8﷯﷮ 4﷮3﷯﷯﷮4﷯﷯ = 1﷮2﷯ 3﷮4﷯﷯ 8﷯﷮ 4﷮3﷯﷯ = 1﷮2﷯ 3﷮4﷯﷯ 2﷮3﷯﷯﷮ 4﷮3﷯﷯ = 1﷮2﷯ 3﷮4﷯﷯ 2﷮4﷯﷯ = 6

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