Problem 6 - Calculus Early Transcendentals

Question

Give the antideriyatives of \(e^{-x}\)

Step-by-Step Solution

Answer

Question: Determine the antiderivative of the function \(e^{-x}\). Answer: The antiderivative of \(e^{-x}\) is \(-e^{-x} + C\), where C is the constant of integration.

Step 1: Integrate the exponential function

To find the antiderivative of \(e^{-x}\), we will use the rule for integrating exponential functions. The general rule for integrating \(e^{kx}\) is: $$ \int e^{kx} dx = \frac{e^{kx}}{k} + C $$ where k is a constant and C is the constant of integration.

Step 2: Apply the integration rule to our function

Applying the rule to our function \(e^{-x}\), we have k = -1. So, the antiderivative will be: $$ \int e^{-x} dx = \frac{e^{-x}}{-1} + C $$

Step 3: Simplify the result

In order to simplify the antiderivative, we can write the fraction as a negative exponent: $$ \int e^{-x} dx = - e^{-x} + C $$ So, the antiderivative of \(e^{-x}\) is \(-e^{-x} + C\).

Key Concepts

Integration of Exponential Functions

Understanding the integration of exponential functions is critical for students tackling calculus. Exponential functions are of the form f(x) = e^{kx}, where e is the base of the natural logarithm and k is a constant. When integrating such functions, it's crucial to recognize that the derivative of e^{kx} is ke^{kx}. Thus, the antiderivative reverses this process.

The general formula for integrating an exponential function is:
\[\begin{equation}\int e^{kx} dx = \frac{e^{kx}}{k} + C\end{equation}\]
where C represents the constant of integration, included since an indefinite integral indicates a family of functions. For \[\begin{equation}\int e^{-x} dx,\end{equation}\]
we set k to -1. Applying the formula, we get the solution:
\[\begin{equation}- e^{-x} + C.\end{equation}\]
This process is a cornerstone in integrating exponential functions, enabling the solution of a wide range of problems involving growth and decay models, among others.

Constant of Integration

The constant of integration is a fundamental concept in calculus, especially when dealing with indefinite integrals. When we find the antiderivative of a function, we are essentially reversing the process of differentiation. However, since differentiation of a constant yields zero, any constant added to the antiderivative will still be a valid solution upon differentiating.

So, when we write:
\[\begin{equation}\int e^{kx} dx = \frac{e^{kx}}{k} + C,\end{equation}\]
the C symbolizes all possible constants that could have been originally added to the function before differentiation occurred. Therefore, C captures the idea that there is not just a single antiderivative, but a family of antiderivatives differing by a constant amount. The inclusion of C in the solution is a pivotal part of expressing the most general form of the antiderivative.

Indefinite Integrals

Indefinite integrals represent the collection of all antiderivatives of a given function. Unlike definite integrals, which compute the area under a curve between two points, indefinite integrals do not have bounds and thus include the constant of integration. They are expressed using the integral sign followed by the function to be integrated and dx, which indicates the variable of integration.

For example:
\[\begin{equation}\int e^{kx} dx\end{equation}\]
is an indefinite integral. Solving this provides the most general antiderivative of the exponential function e^{kx}. Indefinite integrals are crucial for understanding antiderivatives and are widely used in solving differential equations, calculating areas, and in many other areas of mathematics and applied sciences.