Question

In the following exercises, compute each indefinite integral. $$ \int e^{2 x} d x $$

Short answer

The indefinite integral is \( \frac{1}{2} e^{2x} + C \).

Step-by-step solution

Recall the Rule for Integrating Exponential Functions

The integral of an exponential function of the form \( \int e^{ax} \, dx \) is \( \frac{1}{a} e^{ax} + C \), where \( a \) is a constant, and \( C \) is the constant of integration.

Identify Parameters

In the given integral \( \int e^{2x} \, dx \), identify \( a \) as 2 because the exponent of \( e \) is \( 2x \).

Apply the Integration Rule

Substitute \( a = 2 \) into the formula from Step 1 to calculate the indefinite integral: \( \int e^{2x} \, dx = \frac{1}{2} e^{2x} + C \).

Write the Final Answer

The indefinite integral of \( \int e^{2x} \, dx \) is \( \frac{1}{2} e^{2x} + C \). Note that \( C \) represents the constant of integration.

Key concepts

Integration of Exponential Functions

When we talk about the integration of exponential functions, it specifically relates to the process of finding the antiderivative of functions that involve an exponential term, such as \(e^{ax}\). This kind of integration is crucial in mathematics because it allows us to solve a wide range of problems, especially those involving growth processes.

A basic formula exists that we must remember: the integral of \(e^{ax}\) with respect to \(x\) is \(\frac{1}{a} e^{ax} + C\). Here, \(a\) represents a constant and \(C\) is the constant of integration. To compute this, we multiply the exponential by \(\frac{1}{a}\).

It's straightforward when you get used to it, as the derivative of an exponential function with a constant rate is intuitively reversed through integration. The function expands in a way that reflects how exponential growth operates, making this rule particularly important in applications involving dynamic changes, like compound interest or population growth.

Constant of Integration

During the process of computing indefinite integrals, the constant of integration, symbolized as \(C\), is a vital part. When finding antiderivatives, we encounter multiple possibilities: they form a family of functions, all differing by a constant factor.

This constant accounts for any shifts along the y-axis in a function's graph. Essentially, it means that infinitely many antiderivatives could satisfy the derivative function. For instance, our solution \(\frac{1}{2} e^{2x} + C\) could be rewritten with any number in place of \(C\) and still be correct.

• It ensures that all possible antiderivatives are covered.
• It's fundamental for solving differential equations where initial conditions specify \(C\).

So, never forget to include \(C\) when handling indefinite integrals. It's small but mighty!

Exponential Growth and Decay

Understanding exponential growth and decay is fundamental in interpreting exponential functions. These concepts describe processes where quantities increase or decrease at rates proportional to their current value.

Exponential growth refers to situations where something increases rapidly, like population growth or radioactive decay, while decay describes exponential reduction over time, which can be seen in sales depreciation or cooling of a hot substance.

When we integrate exponential functions, we often model these phenomena mathematically. The function \(e^{ax}\) plays a pivotal role:

• Exponential growth occurs when \(a > 0\).
• Exponential decay occurs when \(a < 0\).

In applications, knowing how to integrate these functions helps solve real-world problems. This integration allows us to predict future behavior by analyzing an initial value that alters with time.