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  • Integrate the functions in Exercises 1 to 24. 5

Integrate the functions in Exercises 1 to 24.

    Q5.    \frac{1}{x^{\frac{1}{2}}+ x^\frac{1}{3}}    [Hint: \frac{1}{x^{\frac{1}{2}}+ x^\frac{1}{3}} = \frac{1}{x^\frac{1}{3}(1 + \ x^\frac{1}{6})}, put x = t^6]

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best_answer

Put     x = t^6\ \Rightarrow \ dx = 6t^5dt

We get,

                            \int \frac{1}{x^{\frac{1}{2}}+ x^\frac{1}{3}}dx\ =\ \int \frac{6t^5}{t^3+t^2}dt

or                                                         =\ 6\int \frac{t^3}{1+t}dt

or                                                          =\ 6\int \left \{ (t^2-t+1)-\frac{1}{1+t} \right \}dt

or                                                          =\ 6 \left [ \left ( \frac{t^3}{3} \right ) -\left ( \frac{t^2}{2} \right )+t - \log(1+t) \right ]

Now   put    x = t^6 in the above result :

                                                             =\ 2\sqrt{x} -3x^{\frac{1}{3}}+ 6x^{\frac{1}{6}} - 6 \log \left ( 1-x^\frac{1}{6} \right )\ +\ C

Posted by

Devendra Khairwa

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