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  • By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 16.

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

    Q16.    \int_0^\pi\log(1 +\cos x)dx

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We have                         I\ =\ \int_0^\pi\log(1 +\tan x)dx                                    .....................................................................................(i)

By using the property:- 

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

We get, 

 

 or    

                             I\ =\ \int_0^\pi\log(1 +\cos (\Pi -x))dx                                   

                             

                             

                              I\ =\ \int_0^\pi\log(1 -\cos x)dx                                                ....................................................................(ii)

 

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

  

                          2I\ =\ \int_0^\pi\log(1 +\cos x)dx\ +\ \int_0^\pi\log(1 -\cos x)dx

or                       2I\ =\ \int_0^\pi\log(1 -\cos^2 x)dx\ =\ \int_0^\pi\log \sin^2 xdx

or                        2I\ =\ 2\int_0^\pi\log \sin xdx

or                          I\ =\ \int_0^\pi\log \sin xdx                                                     ........................................................................(iii)

or                           I\ =\ 2\int_0^ \frac{\pi}{2} \log \sin xdx                                                    ........................................................................(iv)

or                            I\ =\ 2\int_0^ \frac{\pi}{2} \log \cos xdx                                          .....................................................................(v)

Adding (iv) and (v) we get,

        

                            I\ =\ -\pi \log2

Posted by

Devendra Khairwa

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