If [·] represents the greatest integer function, then the value of ∫0π2x2−cosxdx is
Step 1: Apply greatest integer function property:
Since x+a,Ifa∈Ithen[x]+a
I=∫0π2x2dx+−cosxdx
Check limit:
0<x<π20<x2<π20<x2≤1and1<x2≤π2[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+∫1π21∵−1≤cosx≤10<x<π20<cosx<1[−cosx]=−1=∫0π2(−1)+∫1π2dx=[−x]0π2+[x]1π2=−π2+π2−1=−1I=1
Hence, the value of ∫0π2([x2]dx+(−cosx))dx=1