Consider the given integral.
I=∫11−cotxdx
I=∫11−cosxsinxdx
I=∫sinxsinx−cosxdx
Let,
sinxsinx−cosx=A(sinx+cosx)+B(sinx−cosx)sinx−cosx
sinxsinx−cosx=(A+B)sinx+(A−B)cosxsinx−cosx
Comparing both the sides, we have
A=12,B=12
Therefore,
I=12∫sinx+cosxsinx−cosxdx+12∫sinx−cosxsinx−cosxdx
I=12∫sinx+cosxsinx−cosxdx+12∫1dx
I=12ln(sinx−cosx)+x2+C
Hence, this is the required value of the integral.