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  • By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 14.

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

    Q14.    \int_0^{2\pi}\cos^5xdx

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We have                      I\ =\ \int_0^{2\pi}\cos^5xdx

It is known that :-

                                    \int_0^{2a}f(x)dx\ =\ 2\int_0^{a}f(x)dx                                           If   f (2a - x) = f(x)

                                                                =\ 0                                                                   If  f (2a - x) =  -  f(x) 

Now,    using the above property                        

                                           \cos^5(\Pi - x)\ =\ - \cos^5x

Therefore,                        I\ =\ 0

Posted by

Devendra Khairwa

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