Ohio Assessments for Educators (OAE) Mathematics Practice Exam 2025 - Free Practice Questions and Study Guide

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Question: 1 / 400

What do you obtain when you integrate csc²(x)dx?

-cot(x)

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

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sec(x)

csc(x)

tan(x)

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Ohio Assessments for Educators (OAE) Mathematics Practice Exam 2025 - Free Practice Questions and Study Guide

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