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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+23tan1(2tanx+13)+c
cotx1+cotx+cotx2xdx
=cotx/sinx1+cosxsinx+cos2nsin2ndxdn
=cosx/sinxsin2x+cosxsinx+cos2xsin2xdn
cosxsinxsin2+cosnsinx+cos2ndx
sincosx1+sinxcosxdn
=1+sinxcosx11sinxcosxdn
=dx11+sinxcosxdx
dx22+sinxdn
x22+2tanx1+tan2xdx
=xsec2x(1+tan2x+tanx)dx
xdt1+t+t2 let tanx=t sec2xdn=dt
xdt(t+12)2+(32)2
=x23tan1(t+1/23/2)+C
I=x23tan1(2tanx+13)+C

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