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Question

Integrate with respect to x: 11+cotx

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Solution

Consider the given integral.


I=11cotxdx


I=11cosxsinxdx


I=sinxsinxcosxdx



Let,


sinxsinxcosx=A(sinx+cosx)+B(sinxcosx)sinxcosx


sinxsinxcosx=(A+B)sinx+(AB)cosxsinxcosx



Comparing both the sides, we have


A=12,B=12



Therefore,


I=12sinx+cosxsinxcosxdx+12sinxcosxsinxcosxdx


I=12sinx+cosxsinxcosxdx+121dx


I=12ln(sinxcosx)+x2+C



Hence, this is the required value of the integral.


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