Problem 32 - Calculus and its applications

Question

Find each integral. $$ \int 2 e^{5 x} d x $$

Step-by-Step Solution

Answer

\( \frac{2}{5} e^{5x} + C \)

Step 1: Identify the Integral Form

The given integral is \( \int 2 e^{5x} \, dx \). This is a basic exponential integral, which generally has the form \( \int a e^{bx} \, dx \). The goal is to integrate the exponential function with respect to \( x \).

Step 2: Identify the Constants

In the integral \( \int 2 e^{5x} \, dx \), the coefficient 2 is a constant factor, and the exponent 5 in \( e^{5x} \) is the rate at which the exponential function grows. We can pull out the constant factor from the integration process.

Step 3: Apply the Exponential Integral Rule

The integral of \( e^{bx} \) with respect to \( x \) is \( \frac{1}{b} e^{bx} + C \), where \( C \) is the constant of integration. Applying this to our integral, the expression becomes \( 2 \times \frac{1}{5} e^{5x} = \frac{2}{5} e^{5x} \).

Step 4: Include the Constant of Integration

The final result of the indefinite integral should include the constant of integration \( C \). Therefore, the integral \( \int 2 e^{5x} \, dx \) evaluates to \( \frac{2}{5} e^{5x} + C \).

Key Concepts

Understanding Exponential Functions

Exponential functions are fundamental in mathematics and appear frequently in various fields such as science, engineering, and even finance. They have the general form \( e^{bx} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. In our integral example \( \int 2 e^{5x} \, dx \), the expression \( e^{5x} \) represents an exponential function. The exponent \( 5x \) means this function grows at a rate that is five times whatever value \( x \) takes on.
You might recognize exponential functions by their characteristic curve that grows rapidly and becomes steeper as \( x \) increases. This rapid increase is precisely what makes exponential functions unique and powerful tools for modeling real-world scenarios such as interest calculations in finance or population growth models.
Some key features of exponential functions include:

• They always yield positive results irrespective of the value of \( x \).
• The derivative of an exponential function \( e^{bx} \) gets you back the same function scaled by its rate of growth, \( b \).

Recognizing the exponential function within an integral allows us to apply specific integration techniques tailored to it.

Exploring Integration Techniques

Integration techniques are methods used to find the antiderivative or integral of a function. When dealing with exponential functions like \( e^{bx} \), the approach involves recognizing patterns and using known integral rules. In this case, we specifically use the rule for exponential integrals: the integral of \( e^{bx} \) is \( \frac{1}{b} e^{bx} \). This technique is straightforward once you identify the exponent's coefficient.
The steps applied in our example involve:

• Identifying constants such as the coefficient of \( e^{bx} \). Here, it is 2, which we can factor out from the integral \( \int 2 e^{5x} \, dx \).
• Using the integral rule for \( e^{bx} \), meaning we multiply \( e^{bx} \) by the reciprocal of its rate of growth \( b \). In this exercise, \( b = 5 \), so we multiply \( e^{5x} \) by \( \frac{1}{5} \).

This method, combined with algebraic manipulation of constants, simplifies the integration process and allows for easy computation of the integral. Integration techniques cater to different kinds of functions, each with unique rules to make the process systematic and efficient.

Concept of the Constant of Integration

When performing indefinite integration, such as with \( \int 2 e^{5x} \, dx \), it is crucial to incorporate the constant of integration, denoted by \( C \). The reason for this is that the process of integration determines a family of functions, each of which differs by a constant.
Therefore, not including \( C \) would imply missing out on an infinite number of possible antiderivatives. Each of those antiderivatives could validate different initial conditions or constraints in applied problem-solving scenarios.
Here's how it works:

• The constant of integration accounts for any vertical shifts of the integral's graph on a plane.
• Without \( C \), the solution would not encompass all potential functions that the antiderivative represents.

In mathematical problems, once specific conditions are given (like a point the function passes through), you can determine the exact value of \( C \). Until such conditions are specified, the integration will always reflect \( +C \) to represent all possible antiderivatives. In summary, it completes the picture, ensuring no mathematical stone is left unturned.