cos^2x*csc^3x=cos^2x*1/sin^3xcos2x⋅csc3x=cos2x⋅1sin3x
= cos^2x/sin^2x*cscx=cot^2x*cscx=(csc^2x-1)*cscxcos2xsin2x⋅cscx=cot2x⋅cscx=(csc2x−1)⋅cscx ....(A)
Hence intcos^2x*csc^3xdx=int(csc^2x-1)*cscxdx∫cos2x⋅csc3xdx=∫(csc2x−1)⋅cscxdx
= intcsc^2xcscxdx-intcscxdx∫csc2xcscxdx−∫cscxdx
Let us process the them separately
First Part - We use integration by parts for intcsc^2xcscxdx∫csc2xcscxdx, considering u=cscxu=cscx and v=-cotxv=−cotx and then du=-cotxcscxdxdu=−cotxcscxdx and dv=csc^2xdxdv=csc2xdx and integrating by parts, as intudv=uv-intvdu∫udv=uv−∫vdu we have
intcscx*csc^2xdx=-cscxcotx-int(-cotx)(-cotxcscxdx)∫cscx⋅csc2xdx=−cscxcotx−∫(−cotx)(−cotxcscxdx)
= -cscxcotx-intcot^2xcscxdx−cscxcotx−∫cot2xcscxdx ....(B)
Second Part intcscxdx=int(cscx(cscx-cotx))/((cscx-cotx))dx∫cscxdx=∫cscx(cscx−cotx)(cscx−cotx)dx
= int(csc²x-cscxcotx)/(cscx-cotx)dx
= int(-cscxcotx+csc²x)/(cscx-cotx)dx
As numerator is differential of denominator this is
= ln(cscx-cotx) ....(C)
Combining (B) and (C)
intcos^2x*csc^3xdx=-cscxcotx-intcot^2xcscxdx+ln(cscx-cotx)
Observe from (A) that cos^2x*csc^3x=cot^2xcscx.
Hence, this becomes
2intcos^2x*csc^3xdx=-cscxcotx+ln(cscx-cotx) and
intcos^2x*csc^3xdx=-1/2cscxcotx+1/2ln(cscx-cotx)+c
