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  • By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 1 0 to pi/2 cos^2x dx

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

    Q1.    \int_0^\frac{\pi}{2}\cos^2 x dx

Answers (1)

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We have                 I\ =\ \int_0^\frac{\pi}{2}\cos^2 x dx                                                    .............................................................(i)

By using 

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

We get :- 

                                     I\ =\ \int_0^\frac{\pi}{2}\cos^2 x dx\ =\ \int_0^\frac{\pi}{2}\cos^2\ (\frac{\pi}{2}- x) dx

or                                                                     

                                                                          I\ =\ \int_0^\frac{\pi}{2}\sin^2 x dx                       ................................................................   (ii)

Adding both (i) and (ii),  we get :-  

                                                      \int_0^\frac{\pi}{2}\cos^2 x dx     +\ \int_0^\frac{\pi}{2}\sin^2 x dx\ =\ 2I          

or                                                   \int_0^\frac{\pi}{2}\ (cos^2 x\ +\ sin^2 x) dx\ =\ 2I

or                                                                                     \int_0^\frac{\pi}{2}1. dx\ =\ 2I           

                                                      

or                                                                                             2I\ =\ \left [ x \right ] ^\frac{\Pi }{2}_0\ =\ \frac{\Pi }{2}

or                                                                                               I\ =\ \frac{\Pi }{4}

Posted by

Devendra Khairwa

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