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Question

If xcotx+1cosec2xx2dx=f(x)+c, where limx0+f(x)=0 and cosec x>0. (where c is a constant of integration), then which of the following is/are incorrect

A
f(1)=1
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B
limx0f(x)|x|tanx does not exist.
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C
f(1)=π21
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D
f(x) is periodic with a period 2π
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Solution

The correct options are
B limx0f(x)|x|tanx does not exist.
C f(1)=π21
D f(x) is periodic with a period 2π
I=xcosx+sinx1(xsinx)2dx=sin1(xsinx)+c
f(x)=sin1(xsinx)f(1)=sin1(sin 1)=1
(as cosec x>0) so x0+
limx0+f(x)|x|tanx=sin1(xsinx)|x|sin x×cosx=sin1(x sin x)x sinx×cosx
=1

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