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Question

If 1+cosec x dx=f(x)+C, where C is a constant of integration, then f(x) is equal to

A
cos1(12sinx)
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B
sin1(12sinx)
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C
2sin1(12sinx)
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D
cos1(1+2sinx)
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Solution

The correct option is A cos1(12sinx)
I=1+cosec x dx
=sinx+1sinx dx
=(1+sinx)(1sinx)sinx(1sinx) dx
=cosx14(12sinx)2 dx
Put 12sinx=t
Then I=dt(12)2t2
=sin1(t1/2)+c
=sin1(12sinx)+c
=cos1(12sinx)π2+c
=cos1(12sinx)+C where C=cπ2

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