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Question

If [·] represents the greatest integer function, then the value of 0π2x2cosxdx is


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Solution

Step 1: Apply greatest integer function property:

Since x+a,IfaIthen[x]+a

I=0π2x2dx+cosxdx

Check limit:

0<x<π20<x2<π20<x21and1<x2π2[x2]=0,and[x2]=1

I=01(0+[cosx])dx+0π21+[cosx]

Step 2:solve the integral part

I=01(0+[cosx])dx+0π21+[cosx]=0π2[cosx]dx+1π211cosx10<x<π20<cosx<1[cosx]=1=0π2(1)+1π2dx=[x]0π2+[x]1π2=π2+π21=1I=1

Hence, the value of 0π2([x2]dx+(cosx))dx=1


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