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Prove that`int_0^x[t]dt=([x]([x]-1))/2+[x](x-[x]),`where [.] denotes the greatest integer function. |
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Answer» Correct Answer - NA Let `x=n+fAAn epsilon` and `0leflt1` `:.[x]=n` `int_(0)^(x)[t]dt=int_(0)^(1)[t]dt+int_(1)^(2)[t]dt+int_(2)^(3)[t]dt+………..+int_(n)^(n+f)[t]dt` `=0+1int_(1)^(2)dt+2int_(2)^(3)dt+………….+n int_(n)^(n+f)dt` `=(2-1)+2(3-2)+……..+n(n+f-n)` `=1+2+3..........+(n-1)+nf` `=((n-1)n)/2+nf` `=([x]([x]-1))/2+[x](x-[x])` [from equation 1] |
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