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  • By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 17.

By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.

    Q17.    \int_0^a \frac{\sqrt x}{\sqrt x + \sqrt{a-x}}dx

Answers (1)

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We have                         I\ =\ \int_0^a \frac{\sqrt x}{\sqrt x + \sqrt{a-x}}dx                                                   ................................................................................(i)

By using, we get 

 

                                \ \int_0^a\ f(x) dx\ =\ \ \int_0^a\ f(a-x) dx

We get,                

                                      I\ =\ \int_0^a \frac{\sqrt x}{\sqrt x + \sqrt{a-x}}dx\ =\ \int_0^a \frac{\sqrt {(a-x)}}{\sqrt {(a-x)} + \sqrt{x}}dx                 .................................................................(ii)

 

Adding (i) and (ii) we get :      

                     

                                     2I\ =\ \int_0^a \frac{\sqrt x\ +\ \sqrt{a-x}}{\sqrt x + \sqrt{a-x}}dx

or                                  2I\ =\ \left [ x \right ]^a_0 = a

or                                    I\ =\ \frac{a}{2}

Posted by

Devendra Khairwa

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