Q

# 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,

$I\ =\ -\pi \log2$

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