If ∫xcotx+1√cosec2x−x2dx=f(x)+c, where limx→0+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
limx→0f(x)|x|tanx does not exist.
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C
f(1)=π2−1
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D
f(x) is periodic with a period 2π
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Solution
The correct options are Blimx→0f(x)|x|tanx does not exist. Cf(1)=π2−1 Df(x) is periodic with a period 2π I=∫xcosx+sinx√1−(xsinx)2dx=sin−1(xsinx)+c f(x)=sin−1(xsinx)⇒f(1)=sin−1(sin1)=1 (as cosec x>0) so x→0+ limx→0+f(x)|x|tanx=sin−1(xsinx)|x|sinx×cosx=sin−1(xsinx)xsinx×cosx =1