Calculus: Early Transcendentals, Chapter 5, 5.4, Section 5.4, Problem 15

You need to evaluate the indefinite integral, such that:
int f(theta)d theta = F(theta) + c
int (theta - csc theta* cot theta)d theta = int theta d theta - int (csc theta* cot theta)d theta
Evaluating integral int theta d theta, using the formula int theta^n d theta = (theta^(n+1))/(n+1) + c , yields:
int theta d theta = (theta^2)/2 + c
int (csc theta* cot theta)d theta = int (1/(sin theta)* (cos theta)/(sin theta)) d theta
You need to use substitution to solve the indefinite integral int (csc theta* cot theta)d theta , such that:
sin theta = t => cos theta d theta = dt
Replacing the variable, yields:
int (dt)/(t^2) = int t^(-2) dt = -1/t + c
Replacing back sin theta for t yields:
int (csc theta* cot theta)d theta = -1/(sin theta) + c
Gathering the results, yields:
int (theta - csc theta* cot theta)d theta = (theta^2)/2 + 1/(sin theta) + c
Hence, evaluating the indefinite integral yields int (theta - csc theta* cot theta)d theta = (theta^2)/2 + 1/(sin theta) + c.