Question

In the following exercises, compute each indefinite integral. $$ \int e^{-3 x} d x $$

Short answer

The indefinite integral is \( -\frac{1}{3} e^{-3x} + C \).

Step-by-step solution

Identify the Integral Form

The integral given is \( \int e^{-3x} \, dx \). This is an exponential function. It's important to recognize that exponential integrals have a standard form for integration, particularly when \( e^{ax} \) is involved.

Recall the Integration Formula

The formula for the integral of \( e^{ax} \) is \( \frac{1}{a} e^{ax} + C \) where \( C \) is the constant of integration. In this case, \( a = -3 \).

Apply the Formula

Using the formula: \( \int e^{-3x} \, dx = \frac{1}{-3} e^{-3x} + C \). This gives: \( -\frac{1}{3} e^{-3x} + C \).

Write the Final Expression

The indefinite integral simplifies to: \( \int e^{-3x} \, dx = -\frac{1}{3} e^{-3x} + C \). The constant \( C \) is included because this problem involves an indefinite integral.

Key concepts

Exponential Function

An exponential function is a type of mathematical function that involves the use of exponents. The base of the exponent is usually the number 'e,' which is approximately equal to 2.71828. This function is often represented in the form \( e^{ax} \), where 'a' and 'x' are variables. The main characteristic of an exponential function is that the rate of change is proportional to the current value of the function.

Exponential functions appear frequently in calculus and mathematical modeling, especially for processes that grow or decay at a constant rate. In this specific exercise, we’re looking at \( e^{-3x} \), which represents exponential decay because the exponent has a negative coefficient (-3).

Understanding how to work with exponential functions is crucial in the process of integration, especially since their unique properties allow for specific integration techniques.

Integration Formula

Integration is the process of finding a function given its derivative, and an integration formula provides a shortcut or a rule to simplify this process. For exponential functions of the form \( e^{ax} \), there's a direct integration formula.

When you integrate \( e^{ax} \), the formula is \( \frac{1}{a} e^{ax} + C \). Here, 'a' is the coefficient of 'x', and 'C' represents the constant of integration.

The key is to recognize that an exponential function's derivative and integral are very closely related. The exponential function maintains its form through differentiation and integration, which is a unique and powerful property to leverage when solving these problems.

Constant of Integration

The constant of integration, denoted as 'C', is an integral part of every indefinite integral. It represents an unknown constant that could have been present in the original function before differentiation.

Since differentiation of a constant results in zero, when we work backward to find the antiderivative, we introduce 'C' to account for any constant that was previously in the function. This ensures that the family of functions that share the same derivative are all included.

Without 'C', the integral expression is incomplete. For example, the solution \( -\frac{1}{3} e^{-3x} + C \) correctly includes 'C' to accommodate any possible constant values in the original function.

Integral Computation

Integral computation is all about applying the integration process to solve for antiderivatives. To compute an indefinite integral like \( \int e^{-3x} \, dx \), it's important to identify the form of the function and apply the appropriate integration method or formula.

In this exercise, recognizing the function \( e^{-3x} \) as an exponential allows us to utilize the integration formula for \( e^{ax} \). Substituting \( a = -3 \) into the formula \( \frac{1}{a} e^{ax} \) gives us \( -\frac{1}{3} e^{-3x} \).

The integration result includes a constant 'C', ensuring that we cover all potential solutions. Integral computation thus involves recognizing the form of the function, correctly applying known formulas, and always including constants where appropriate.