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int(e^(cotx))/(sin^(2)x)("2 ln cosec x"+...

`int(e^(cotx))/(sin^(2)x)("2 ln cosec x"+sin2x)dx`

A

`2e^(cotx)ln |sinx|+c`

B

`2e^(tanx)ln|sinx|+c`

C

`2e^(cotx)ln|cosx|+c`

D

`2e^(tanx)ln |cosx|+c`

Text Solution

Verified by Experts

The correct Answer is:
A
`t=cotxrArrdt=-"cosec"^(2)xdx`
`I=int(e^(cotx))/(sin^(2)x)(ln" cosec"^(2)x+(2tanx)/(1+tan^(2)x))dx`
`=-inte^(t)(ln(1+t^(2))+((2)/(t))/(1+(1)/t^(2)))dt`
`=-e^(t)ln|1+t^(2)|+c=-2e^(cotx)ln|"cosec x"|+c`
`=2e^(cotx)ln|sinx|+c`
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