Split the integral up into two
#int cos^2(x)sin(x)-tan^2(x)cot(x) dx = int cos^2(x)sin(x)dx - int tan^2(x)cot(x) dx#
First Step
For #int cos^2(x)sin(x)dx#,
Let #u=cos(x)# and thus #du=-sin(x)dx#
Substituting, you get
#int cos^2(x)sin(x)dx=-int cos^2(x)(-sin(x)dx)=-int u^2du=-u^3/3+C =-cos^3(x)/3+C#
Second Step
For #int tan^2(x)cot(x) dx#,
Since #tan^2(x)cot(x)=tan^2(x)/tan(x)=tan(x)#
#int tan^2(x)cot(x)=int tan(x)dx#
At this point, you can use a formula sheet to get the answer directly, but if you are interested, you can follow along the next few steps.
#int tan(x)dx = int sin(x)/cos(x) dx#
Now if you let #w=cos(x)#, then #dw=-sin(x)dx#
#int sin(x)/cos(x) dx = -int 1/cos(x) * -sin(x)dx=-int 1/w dw = -lnabs(w)+C=lnabs(cos(x))+C#
Final step
Hence, by subtracting the two integrals, one gets
#int cos^2(x)sin(x)-tan^2(x)cot(x) dx = -cos^3(x)/x - (-lnabs(cos(x)))+C = lnabs(cos(x))-cos^3(x)/x+C#
