Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the result of integrating sec²(x)dx?

1. sec(x)

2. -tan(x)

3. tan(x)

4. csc(x)

The correct answer is: tan(x)

The integral of sec²(x) with respect to x can be directly related to a well-known differentiation rule. The derivative of tangent function, tan(x), is sec²(x). This means that when you integrate sec²(x), you are essentially asking for the function whose derivative returns sec²(x). Thus, the integral of sec²(x) is tan(x), plus a constant of integration, which is typically represented as C. The other potential answers do not apply in this case. The function sec(x) is the secant function, which is not the integral of sec²(x). Similarly, -tan(x) represents the negative of the tangent function, which would be derived from the integral of -sec²(x) dx, and csc(x), the cosecant function, is unrelated to the integral of sec²(x) altogether. Therefore, the correct response to this integration problem is indeed tan(x).