Problem 320 - Calculus Volume 1

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Integral Form

The given integral is \( \int e^{2x} \, dx \). This integral involves an exponential function with a linear expression in the exponent.

Step 2: Apply the Integration Rule for Exponentials

Use the rule for integrating exponential functions, which states that \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \), where \( a \) is a constant and \( C \) is the constant of integration.

Step 3: Substitute the Appropriate Values

For the integral \( \int e^{2x} \, dx \), the constant \( a = 2 \). Applying the rule gives us \( \frac{1}{2} e^{2x} + C \).

Step 4: Write the Final Solution

The indefinite integral of \( e^{2x} \) is \( \frac{1}{2} e^{2x} + C \), where \( C \) is the constant of integration.

Key Concepts

Exponential Functions

Exponential functions are mathematical functions of the form \( f(x) = a \cdot e^{bx} \), where \( e \) is the base of the natural logarithms, approximately equal to 2.71828, and \( a \) and \( b \) are constants. These functions exhibit constant proportional growth or decay, which is why they're commonly used in modeling population growth, radioactive decay, and interest calculations.
When we encounter exponential functions in calculus, especially in integration, understanding their properties is crucial. For example, the derivative of \( e^{x} \) with respect to \( x \) is itself \( e^{x} \). This unique property simplifies various calculus operations, including integration.
For the function \( e^{bx} \), where \( b \) is a constant, the process of differentiation and integration becomes straightforward due to this consistent behavior. Hence, while computing the indefinite integral of an exponential function, we can apply specific rules to obtain the solution efficiently.

Integration Rules

Integration rules are formulas and techniques that allow us to find the integral of a function. Integrating functions is one of the fundamental operations in calculus, enabling the calculation of areas under curves and solutions to differential equations.
A critical rule for exponential functions of the form \( e^{ax} \) is that the integral is \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \), where \( C \) represents the constant of integration. The constant \( a \) in the exponent modifies the scaling factor of the integral.
This formula is derived from the understanding of how exponential functions differ and integrate. By recognizing that integration is the inverse operation of differentiation, we see that this rule efficiently allows us to compute the indefinite integral of exponentials.Thus, when dealing with exponential integrals, it's essential to identify the \( a \) and apply the rule directly.

Integration Techniques

Mastering integration techniques involves learning various methods to handle different types of functions beyond standard rules. These techniques include substitution, integration by parts, and partial fraction decomposition.
For the integral \( \int e^{2x} \, dx \), we used the direct application of the exponential integration rule. But in more complex cases, such as \( \int e^{f(x)} f'(x) \, dx \), substitution might be necessary. Here you would set \( u = f(x) \), then find \( du \), and rewrite the integral in terms of \( u \) to simplify.

• Substitution: Particularly useful for integrals of composite functions.
• Integration by Parts: Used for products of functions, based on the product rule for differentiation.
• Partial Fractions: Helpful for rational functions, breaking complex fractions into simpler parts.
  Using these techniques effectively allows for tackling a variety of integrals that a straightforward application of rules couldn't solve, ensuring a comprehensive approach to integral calculus.