Question

# Solve: ∫cotx1+cotx+cot2xdx

A
x+23tan1(2tanx+13)+c
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B
x+23cot1(2cotx+13)+c
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C
x+32cot1(2tanx+13)+c
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D
x2+23tan1(2cotx+13)+c
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Solution

## The correct option is A x+2√3tan−1(2tanx+1√3)+c∫cotx1+cotx+cotx2xdx=∫cotx/sinx1+cosxsinx+cos2nsin2ndxdn=∫cosx/sinxsin2x+cosxsinx+cos2xsin2xdn∫cosxsinxsin2+cosnsinx+cos2ndx∫sincosx1+sinxcosxdn=∫1+sinxcosx−11−sinxcosxdn=∫dx−∫11+sinxcosxdx∫dx−∫22+sinxdnx−∫22+2tanx1+tan2xdx=x−∫sec2x(1+tan2x+tanx)dxx−∫dt1+t+t2 let tanx=t sec2xdn=dtx−∫dt(t+12)2+(√32)2=x−2√3tan−1(t+1/2√3/2)+CI=x2√3tan−1(2tanx+1√3)+C

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