Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the integral of csc(x)cot(x)dx?

1. -csc(x)

2. csc(x)

3. sin(x)

4. cot(x)

The correct answer is: -csc(x)

The correct answer to the integral of csc(x)cot(x)dx is indeed -csc(x). To understand why this is the case, we first need to recall the definitions of the trigonometric functions involved. The cosecant function, csc(x), is defined as 1/sin(x), and cotangent, cot(x), is defined as cos(x)/sin(x). Therefore, the product csc(x)cot(x) can be rewritten as (1/sin(x))(cos(x)/sin(x)), which simplifies to cos(x)/sin^2(x). When integrating csc(x)cot(x), we can utilize a substitution approach. Notice that the derivative of csc(x) is -csc(x)cot(x). This relationship shows that when we integrate csc(x)cot(x) with respect to x, we are essentially reversing the differentiation process for csc(x). Integrating csc(x)cot(x)dx, we get -csc(x) + C, where C is the constant of integration. Thus, the integral directly leads to -csc(x), confirming that this is the correct result. Other options, such as csc(x), sin(x), and cot(x), either