Problem 59 - Calculus Early Transcendentals

Question

Evaluate the following integrals. $$\int 3^{-2 x} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the integral of the given function: $$\int 3^{-2x} dx$$ Answer: The integral of the given function is $$\int 3^{-2x} dx = \frac{1}{2\ln{3}} 3^{-2x} + C$$

Step 1: Identify the function type

We are given a function in the form of an exponential function. The integral we need to evaluate is $$\int 3^{-2 x} d x.$$

Step 2: Rewrite using exponent properties

We can rewrite the integral in the form of a more familiar exponential function using the property \(a^{-b} = \frac{1}{a^b}\). So the integral becomes: $$\int \frac{1}{3^{2 x}} dx$$

Step 3: Integrate

In order to integrate the function, we can use the formula for the integration of exponential functions: $$\int a^{kx}dx = \frac{1}{k\ln{a}} a^{kx} + C$$ where \(a\) is the base, \(k\) is the coefficient in the exponent, and \(C\) is the constant of integration. In our case, \(a = 3\) and \(k = 2\). Therefore, the integral becomes: $$\int \frac{1}{3^{2 x}} dx = \frac{1}{2\ln{3}} 3^{-2 x} + C$$

Step 4: Simplify the result

Now we have the result in the form of an exponential function. We can simplify it by writing it back in the original form: $$\frac{1}{2\ln{3}} 3^{-2 x} + C = \frac{1}{2\ln{3}} 3^{-2 x} + C$$ Therefore, the final answer for the given integral is: $$\int 3^{-2 x} d x = \frac{1}{2\ln{3}} 3^{-2 x} + C$$

Key Concepts

Integral Calculus

Integral calculus is a fundamental branch of calculus involving the concept of an integral. It mainly focuses on finding the area under curves or accumulating quantities, such as the total distance from velocity over time.
In the context of calculus, integration is viewed as the reverse operation to differentiation. Its main purpose is to measure how functions accumulate values and how they can generate areas or volumes. This is known as the "Fundamental Theorem of Calculus." This theorem connects differentiation and integration, two opposite operations.

• Definite Integral: Calculates the total accumulation between two specified limits.
• Indefinite Integral: Represents a family of functions and contains a constant of integration often written as 'C'.

In practical terms, solving an integral involves finding a function whose derivative matches the original function given in the problem. For example, finding the integral of a simple function is like asking, "From what function does this one arise if it was derived?" Understanding this gives one a handle on how quantities change and accumulate.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions appear frequently in various scientific fields.
Such functions can be identified by their general form: \[ y = a^{x} \]where 'a' is the base and 'x' is the exponent. In calculus, handling exponential functions often involves laws of exponentiation, allowing one to simplify and manipulate expressions.
Integration of exponential functions requires understanding their unique derivative properties. The exponential function's rate of change is proportional to its current value, making it a popular model for growth and decay phenomena, such as population growth and radioactive decay.
In our example, we evaluated \[ \ \int 3^{-2x} dx \]After applying integration techniques, it demonstrates how an exponential decay function can be integrated successfully using its unique properties.

Integration Techniques

Integration techniques are methods used to find integrals, especially when dealing with complex expressions. A good grasp of these techniques enables solving a wide variety of problems.

• Substitution Method: Used when an integral contains a composite function, simplifying it to a more manageable form.
• Integration by Parts: Solutions for integrals of a product of two functions, based on the product rule of differentiation.
• Partial Fractions: Breaking down complex rational functions into simpler fractions that are easier to integrate.

Integrating exponential functions like \[ \ \int a^{kx} dx \]requires applying a specific formula, \[ \ \frac{1}{k \ln a} a^{kx} + C \]Understanding when and how to apply such formulas is essential for solving complex exponential integrations. Mastery of these techniques ensures efficient problem-solving, close attention in applying the right technique to each situation, verifies the accuracy of results. Every technique has its domain for application, making the seamless navigation through various integration problems achievable.
Ultimately, integration techniques are your toolbox for transforming complex calculus problems into solvable parts.