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Question

dxcotx2cotx3cotx6 =

A
lnsecx2secx3secx6+c
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B
lnsec2x2sec3x3sec6x6+c
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C
lnsec2x2cos3x3cos6x6+c
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D
lntan2x2tan3x3sec5x6+c
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Solution

The correct option is C lnsec2x2cos3x3cos6x6+c
dxcotx2cotx3cotx6
I=tanx2tanx3tanx6dn
tanx6=tan(x2x3) [tan(AB)=tanAtanB1+tanA+tanB]
tanx6=tanx/2+tanx/31+tanx/2tanx/3
tanx6(1+tanx2tanx3)=tanx2tanx3
tanx6+tanx2tanx3tanx6=tanx2tanx3
Integrating both sides we get
tanx6dx+tanx2tanx3tanx6dn=tanx2dxtanx3dn
I=tanx2dxtanx3dntanx6dn [from equation]=2logsecx23logsecx36logsecx6+C
=logsec2x2+logcos3x3+logcos6x6+C
I=logsec2x2+cos3x3+cos6x6+C

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