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If `int_(0)^(x)[x]dx=int_(0)^(|x|)x dx,` (where [.] and {.} denotes the greatest integer and fractional part respectively), thenA. `x epsilon[0,1)`B. `{x}=1//2`C. `{x}=1//3`D. `xgt0` |
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Answer» Correct Answer - A::B `int_(0)^(x)[x]dx=int_(0)^(1)0dx+int_(1)^(2)1dx+int_(2)^(3)2dx+…………….` `+int_([x]-1)^([x])([x]-1)dx+int_([x])^(x)[x]dx` `=0+1+2+3+……….+([x]-1)+[x](x-[x])` `=(([x]-1)[x])/2+[x]{x}`…………..1 Now `int_(0)^([x]) dx=[(x^(2))/2]_(0)^([x])=([x]^(2))/2`..............2 Comparing 1 and 2 we get `([x]-1)[x])/2+[x]{x}=([x]^(2))/2` `:.[x]^(2)-[x]+2[x]{x}=[x]^(2)` `=[x]=0` or `{x}=1//2` |
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