Ohio Assessments for Educators (OAE) Mathematics Practice Exam

When integrating the function \( \csc^2(x) \), the result is derived from the relationship between the trigonometric functions involved. The integral of \( \csc^2(x) \) is specifically tied to the derivative of \( \cot(x) \). The derivative of \( \cot(x) \) is \( -\csc^2(x) \). Therefore, when you perform the integration of \( \csc^2(x) \), you're essentially reversing the differentiation process, which introduces a negative sign. This leads to the integral \( \int \csc^2(x) \, dx = -\cot(x) + C \), where \( C \) is the constant of integration. The other choices relate to different trigonometric integrals: the integral of \( sec^2(x) \) yields \( tan(x) \), the integral of \( csc(x) \) yields \( -\ln |csc(x) + cot(x)| + C \), and none of these integrations yield \( -\cot(x) \). Thus, the correct integral of \( \csc^2(x) \) supports the choice of \( -\cot(x) \) as