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`int_(0)^(pi)[cotx]dx,` where [.] denotes the greatest integer function, is equal toA. `(pi)/2`B. `1`C. `-1`D. `-(pi)/2` |
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Answer» Correct Answer - D Let `I=int_(0)^(pi)[cotx]dx` …………..i `=int_(0)^(pi)[cot(pi-x)]dx=int_(0)^(pi)[-cotx]dx`………….ii Adding i and ii we get `2I=int_(0)^(pi)[cotx] dx+int_(0)^(pi)[-cotx]dx=int_(0)^(pi)(-1)dx` [since `[x]+[-x]` is equal to `-1` if `x !inZ` and is equal to 0 if `x epsilonZ`] `=[-x]_(0)^(pi)=-pi` `:.I=-(pi)/2` |
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