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mustafa bandook wala Grade: 12
        

 definite integration of sin^2 x/sin x+cos x from 0 to pie

7 years ago

Answers : (1)

vikas askiitian expert
510 Points
										

integral sin^2x/sinx+cosx from 0 to pie =


 


 integral{( sin^2x)/(sinx+cosx)  + (cos^2x)/(sinx+cosx)}from 0to pie/2     after applying propertie of integral


 


=integral( 1/sinx+cosx)integral from 0 t0 pie/2


 


=  put sinx=2tan(x/2)/(1+tan^x/2) and cosx=(1-tan^x/2)/(1+tan^2x/2)


 


now expression becomes   integral (sex^2x/2) /(1+2tan(x/2) -tan^2x/2) limit from 0 to pie/2 


 


now put tanx/2 = t


    we get sec^2dx=2dt 


expression becomes integral 2dt/1+2t-t^2  limit 0 to 1


 =integral -2/t^2 - 2 limit 0 to 1


 =(1/underoot2) .(log(underoot2- 1)/(underoot2 +1))

7 years ago
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