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  • By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 12

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

    Q12.    \int_0^\pi\frac{xdx}{1+\sin x}

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We have                   I\ =\ \int_0^\pi\frac{xdx}{1+\sin x}                                           ..........................................................................(i)

By using the identity :-  

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

We get,   

                                I\ =\ \int_0^\pi\frac{xdx}{1+\sin x}\ =\ \int_0^\pi\frac{(\Pi -x)dx}{1+\sin (\Pi -x)}

or                            I\ =\ \int_0^\pi\frac{(\Pi -x)dx}{1+\sin x}                                         ............................................................................(ii)

 

Adding both (i) and (ii) we get,     

                        

                               2I\ =\ \int_0^\pi\frac{\Pi}{1+\sin x} dx

or                             2I\ =\ \Pi \int_0^\pi\frac{1-\sin x}{(1+\sin x)(1-\sin x)} dx\ =\ \Pi \int_0^\pi\frac{1-\sin x}{\cos^2 x} dx

or                             2I\ =\ \Pi \int_0^\pi (\sec^2\ -\ \tan x \sec x) x dx

or                               I\ =\ \Pi

 

 

Posted by

Devendra Khairwa

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