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  • Integrate the functions in Exercises 1 to 24. 2

Integrate the functions in Exercises 1 to 24.

    Q2.    \frac{1}{\sqrt{x+a} + \sqrt{x+b}}

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At first we will simplify the given expression,

                               \frac{1}{\sqrt{x+a} + \sqrt{x+b}}\ =\ \frac{1}{\sqrt{x+a} + \sqrt{x+b}}\times\frac{\sqrt{x+a} - \sqrt{x+b}}{\sqrt{x+a} - \sqrt{x+b}}

or                                                                     =\ \frac{\sqrt{x+a} - \sqrt{x+b}}{a-b}

Now taking its integral we get,

                                               \int \frac{1}{\sqrt{x+a} + \sqrt{x+b}}\ =\ \frac{1}{a-b}\int (\sqrt{x+a} -\sqrt{x+b})dx

or                                                                                          =\ \frac{1}{a-b}\left [ \frac{(x+a)^{\frac{3}{2}}} {\frac{3}{2}}\ -\ \frac{(x+b)^{\frac{3}{2}}} {\frac{3}{2}} \right ]

or                                                                                          =\ \frac{2}{3(a-b)}\left [ (x+a)^{\frac{3}{2}}\ -\ (x+b)^{\frac{3}{2}} \right ]\ +\ C

Posted by

Devendra Khairwa

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