Problem 29 - Calculus and its applications

Question

Find each integral. $$ \int e^{7 x} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{1}{7}e^{7x} + C \).

Step 1: Identify the Function and Basic Rule

We start with the integral \( \int e^{7x} \, dx \). Recognize that this is an exponential function. The rule for integrating \( e^{kx} \) is \( \frac{1}{k}e^{kx} + C \).

Step 2: Apply the Exponential Rule

Using the rule from Step 1, identify \( k \) as 7. Apply the integration rule: \( \int e^{7x} \, dx = \frac{1}{7}e^{7x} + C \).

Step 3: Simplify the Expression

The expression is already simplified, so the final answer is \( \frac{1}{7}e^{7x} + C \), where \( C \) is the constant of integration.

Key Concepts

Exponential Function

An exponential function is one where a constant base is raised to a power that is a linear function of the variable. In simpler terms, it looks like \( a^{fx} \), where \( a \) is a positive constant and \( fx \) includes the variable. When we see functions like \( e^{7x} \), it means we've got the base \( e \), a special number approximately equal to 2.718, which is called the natural exponential base. This base is very important in calculus because it arises naturally in many growth processes. Exponential functions grow very quickly as the variable increases, making them essential to model processes like population growth, radioactive decay, or even certain financial models. When integrated, these functions reveal accumulated quantities over time.

Integration Rule

Integration is essentially the opposite operation of differentiation. It involves finding a function that represents the area under a curve. For exponential functions like \( e^{kx} \), where \( k \) is a constant, there's a specific rule we use to integrate it easily:

• Identify the constant \( k \) in the exponent.
• Use the integration formula \( \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \).

This rule emerges from the fact that the derivative of \( e^{kx} \) is \( ke^{kx} \), so integrating reverses this differentiation effect. By applying this rule, we simplify the potentially complex process of integration into a straightforward calculation. This rule allows us to quickly determine the 'antiderivative' of these exponential functions.

Constant of Integration

When we integrate a function, we add a constant \( C \) known as the constant of integration. This arises because integration is the reverse of differentiation, and when we differentiate any constant, it becomes zero. Thus, when reversing that process, we need to account for any constant that might have existed. This is essential because every time we integrate, the solution is not unique – it represents a family of functions that all share the same rate of change but are vertically shifted. In practical terms:

• Whenever you integrate a function, always remember to add \( + C \) at the end.
• This accounts for any constant that was "lost" during differentiation.
• It reflects the idea of general solutions in calculus, showing all possible functions that could give the same derivative.

Understanding the constant of integration is crucial for solving many problems in calculus, especially those involving initial conditions that allow us to find a specific solution from the family of solutions given by the integral.