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  • Show that integral 0 to a f(x)g(x)dx = 2 integral 0 to a f(x) dx, if f and g are defined as f(x) = f(a - x) and g(x) + g( a - x) = 4

Q19.    Show that \int_0^a f(x)g(x)dx = 2\int_0^af(x)dx if f and g are defined as f(x) = f(a-x) and g(x) + g(a-x) = 4

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Let                        I\ =\ \int_0^a f(x)g(x)dx                                                                              ........................................................(i)

This can also be written as    :

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

or                                                     I\ =\ \int_0^a f(x)g(a-x)dx                                                 ................................................................(ii)

Adding (i) and (ii), we get,              

                                                      2I\ =\ \int_0^a f(x)g(a-x)dx +\ \int_0^a f(x)g(x)dx

                                                        2I\ =\ \int_0^a f(x)4dx

or                                                       I\ =\ 2\int_0^a f(x)dx

 

 

 

Posted by

Devendra Khairwa

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