∫e^x dx

Definition

The integral of the exponential function e^x with respect to the variable x. This represents the antiderivative or indefinite integral of the exponential function, which is a fundamental operation in calculus involving exponential and logarithmic functions.

5 Must Know Facts For Your Next Test

1. The indefinite integral of the exponential function e^x is the function $e^x + C$, where $C$ is an arbitrary constant of integration.
2. The derivative of the exponential function e^x is e^x, so the antiderivative or integral of e^x is also e^x.
3. The integral $\int e^x dx$ represents the area under the curve of the exponential function e^x on the interval [a, b].
4. The integral $\int e^x dx$ is a fundamental operation in calculus and appears in many applications, such as modeling exponential growth and decay processes.
5. The value of the integral $\int e^x dx$ depends on the limits of integration, and the constant of integration $C$ represents the family of all antiderivatives of the exponential function.

Review Questions

• Explain the relationship between the exponential function e^x and its indefinite integral ∫e^x dx.
  • The indefinite integral of the exponential function e^x is the function e^x + C, where C is an arbitrary constant of integration. This is because the derivative of the exponential function e^x is e^x itself. Therefore, the integral ∫e^x dx represents the antiderivative or primitive function of the exponential function, which is a fundamental operation in calculus involving exponential and logarithmic functions.
• Describe the geometric interpretation of the integral ∫e^x dx.
  • The integral ∫e^x dx represents the area under the curve of the exponential function e^x on a given interval [a, b]. This area can be calculated using the fundamental theorem of calculus, which states that the integral of a function f(x) over an interval [a, b] is equal to the difference between the values of the antiderivative F(x) at the endpoints of the interval, i.e., F(b) - F(a), where F(x) is any function whose derivative is f(x).
• Discuss the applications of the integral ∫e^x dx in modeling real-world phenomena.
  • The integral ∫e^x dx is a fundamental operation in calculus and has numerous applications in modeling real-world phenomena. For example, it can be used to model exponential growth and decay processes, such as the growth of a population, the decay of radioactive materials, or the charging and discharging of capacitors in electrical circuits. Additionally, the integral ∫e^x dx appears in the solutions of differential equations that describe various physical, biological, and engineering systems involving exponential functions.

Related terms

Indefinite Integral: The set of all antiderivatives or primitive functions of a given function. The indefinite integral is denoted with the symbol ∫ and represents the general solution to the problem of finding a function whose derivative is the given function.

Antiderivative:

A function whose derivative is the original function. The antiderivative of a function f(x) is a function F(x) such that F'(x) = f(x).

Exponential Function: A function of the form f(x) = a^x, where a is a positive constant. The exponential function e^x is a special case where a = e, the base of the natural logarithm.