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Question

Integral of 1+2cotx(cotx+ cosecx) w.r.t x is :

A
2lncosx2+c
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B
2lnsinx2+c
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C
12lncosx2+c
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D
lnsinxln( cosecxcotx)+c
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Solution

The correct option is B 2lnsinx2+c
Given , 1+2cotx(cotx+ cosecx)
1+2cot2x+2cotx cosecx
1+2cos2xsin2x+2cosxsin2x
sin2xsin2x+2cos2xsin2x+2cosxsin2x
sin2x+2cos2x+2cosxsinx
1cos2x+2cos2x+2cosxsinx
cos2x+2cosx+1sinx
(cosx+1)2sinx
(cosx+1)sinx
writing sinx and cosx in submultiple angle we have,
(2cos2x21+1)2sinx2cosx2
(2cosx2)2sinx2
cotx2
Since , cotx=ln|sinx|+c
So, cotx2=2ln|sinx2|+c
Hence, option 'B' is correct.

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