area bounded by y=[sin x+cos x] and the lines x=0,x=pi is (where [.] denotes the greatest integer function)

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We havey=sinx+cosxRequired area will be given by:A=0πy.dx=0πsinx+cosx.dx=0π2sinx.12+cosx.12.dx=0π2sinx.cosπ4+cosx.sinπ4.dx=0π2sinx+π4.dx  sinA.cosB+cosA.sinB=sinA+BAs 0<x<π0+π4<x+π4<π+π4π4<x+π4<5π4When0<xπ2π4<x+π43π4Hence 2sinx+π4[1,2)2sinx+π4=1When π2<x3π43π4<x+π4π Hence 2sinx+π4[0,1)2sinx+π4=0When3π4<x<ππ<x+π4<5π4Hence 2sinx+π4-1,02sinx+π4=-1Now0π2sinx+π4.dx=0π22sinx+π4.dx+π23π42sinx+π4.dx+3π4π2sinx+π4.dx=0π21.dx+π23π40.dx+3π4π-1.dx=0π21.dx-3π4π1.dx=x0π2-x3π4π=π2-0-π-3π4=π2-π4=π40πsinx+cosx.dx=π4

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