Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the integral of e^x?

1. e^x + C

2. ln(e^x) + C

3. 1/e^x + C

4. 0

The correct answer is: e^x + C

The integral of \( e^x \) is \( e^x + C \), where \( C \) represents the constant of integration. This result stems from the fundamental property of the exponential function. The unique quality of the function \( e^x \) is that it is its own derivative, meaning that when you differentiate \( e^x \), you get \( e^x \) back. Therefore, when you integrate \( e^x \), the result is also \( e^x \), plus the constant \( C \) to account for the family of functions that differ by a constant. The other options do not represent the correct integral of \( e^x \). The choice involving \( \ln(e^x) + C \) complicates the expression unnecessarily, as the integral does not involve logarithms in this context. The option \( 1/e^x + C \) suggests an integration of a reciprocal function, which is incorrect for \( e^x \). Finally, the option of \( 0 \) is incorrect because the integral of a non-zero function cannot be zero in this case, as \( e^x \) is always positive. Therefore, the integral \( e^x + C \