Calculus of a Single Variable, Chapter 4, 4.1, Section 4.1, Problem 27

int(1-csc(t)cot(t)dt
=intcotdt-intcsc(t)cot(t)dt
=int(cos(t)/sin(t))dt-int(1/sin(t)(cos(t)/sin(t))dt
=intcos(t)/sin(t)dt-intcos(t)/(sin^2(t))dt
Now,
intcos(t)/sin(t)dt
let x=sin(t)
dx=cos(t)dt
intdx/x
=ln(x)
=ln(sin(t)
intcos(t)/(sin^2(t))dt
Let sin(t)=y
dy=cos(t)dt
=int(1/y2)dy
=y^(-2+1)/(-2+1)
=-1/y
=-1/sin(t)
:.int(1-csc(t)cot(t)dt=ln(sin(t))-(-1/sin(t))+C
C is a constant
=ln(sin(t))+1/sin(t)+C