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Question

Integrate the following functions.
1(1+cotx)dx.

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Solution

Let I=1(1+cotx)dx=11+cosxsinxdx
=1sinx+cosxsinxdx=sinx(sinx+cosx)dx=122(sinx)sinx+cosxdx=12sinx+sinx+cosxcosxsinx+cosxdx
[add and subtract cos x in numerator]
=12(sinx+cosx)+(sinxcosx)(sinx+cosx)dx=12[(sinx+cosx)(sinx+cosx)dx+(sinxcosx)sinx+cosxdx]=12[1dx+(sinxcosx)sinx+cosxdx]
Let sinx+cosx=tcosxsinx=dtdx
dx=dt[sinxcosx]I=12[1dx+(sinxcosx)tdt[sinxcosx]]=12[1dx1tdt]=12[xlog|t|]+C=12[xlog|sinx+cosx|]+C


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