Problem 12 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int e^{-2 x} d x $$

Step-by-Step Solution

Answer

\(\int e^{-2x} dx = \frac{1}{-2} e^{-2x} + C\)

Step 1: Identify the function to be integrated

The function to be integrated is \(e^{-2x}\).

Step 2: Apply the formula for integrating exponential functions

Use the formula \(\int e^{ax} dx = \frac{1}{a} e^{ax} + C\), where \(a\) equals to -2 in this case. That is, \(\int e^{-2x} dx = \frac{1}{-2} e^{-2x}\).

Step 3: Add the constant of integration

Every time an indefinite integral is computed, a constant of integration should be added to the result. Thus, \(\int e^{-2x} dx = \frac{1}{-2} e^{-2x} + C\).

Key Concepts

Integration

When dealing with the mathematical concept of integration, it's essential to understand its significance in relation to finding an antiderivative or the area under a curve. Integration is essentially the opposite operation of differentiation. While differentiation gives the rate at which a quantity changes, integration provides the accumulation of quantities.

For example, in physics, if you know the speed of an object at any given time, integration can be used to find the object's position from its speed by accumulating the changes over time. When solving an indefinite integral, such as \(\int e^{-2 x} dx\), the goal is to find a function that, when differentiated, would result in the original function we are integrating, in this case \(e^{-2x}\).

The process does not determine a single specific function, but rather a family of functions that share the same derivative. This is why the answer includes a constant of integration, which represents an unknown constant that could be any real number. Integration is a fundamental tool in calculus that allows us to find solutions to a vast array of problems in engineering, science, economics, and beyond.

Exponential Functions

Exponential functions, symbolized by the mathematical expression \(e^{ax}\), where \(e\) is the base of the natural logarithm and \(a\) is a constant, are vital in various fields of science and mathematics. These functions describe growth or decay processes, such as population growth, radioactive decay, and interest compounding.

What makes exponential functions particularly interesting is that their rate of growth (or decay) is proportional to their current value. This characteristic yields a constant rate of change, which is why their derivative is directly proportional to the function itself; for \(e^{ax}\), the derivative is \(ae^{ax}\).

When integrating an exponential function, understanding this relationship between the function and its derivative simplifies the process immensely. As observed in the exercise \(\int e^{-2 x} dx\), the pre-existing knowledge about the nature of exponential growth and decay allows for direct application of integration rules tailored for these functions, showcasing the profound link between the concepts of integration and exponential functions.

Constant of Integration

The constant of integration, denoted as \(C\), plays a crucial role in indefinite integrals. Since an indefinite integral represents a family of functions with the same derivative, the constant of integration embodies the vertical shift that each member of this family can have.

When we integrate a function to find its antiderivative, there are potentially infinite functions satisfying the condition—each differs by a constant amount. This is because derivatives of constant terms are zero, meaning that during the differentiation process, information about any constant term is lost. Upon reversal through integration, we must account for that loss by including \(C\) in our final expression.

Therefore, when solving \(\int e^{-2 x} dx\), as depicted in the provided steps, we append \(C\) to the result to acknowledge all the possible original functions that would yield \(e^{-2x}\) as their derivative. The constant of integration is fundamental to the understanding of indefinite integrals and is essential in ensuring that our solutions are complete and general, covering all possible scenarios.