Final answer:
The integration of -csc(x)tan(x) results in ln|sec(x) + tan(x)| + C, where C is the constant of integration.
Step-by-step explanation:
The question asks us to find the integral of the function -csc(x)tan(x). When we look at the integral of -csc(x)tan(x), we can simplify our work by recognizing that csc(x) is 1/sin(x) and tan(x) is sin(x)/cos(x). Therefore, when we multiply -csc(x) by tan(x), we get -sin(x)/(sin(x)cos(x)), which simplifies to -1/cos(x), or -sec(x).
Now, the integral of -sec(x) is ln|sec(x) + tan(x)| + C, where C is the constant of integration. This result comes from a standard integral that is often memorized or found in a table of integrals. So, the integral of -csc(x)tan(x) is ln|sec(x) + tan(x)| + C.