Problem 52 - University Calculus: Early Transcendentals

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(2 e^{x}-3 e^{-2 x}\right) d x$$

Step-by-Step Solution

Answer

The most general antiderivative is \( 2e^x + \frac{3}{2}e^{-2x} + C \).

Step 1: Identify the integrals

The given integral is \( \int (2e^x - 3e^{-2x}) \, dx \). We need to integrate each term separately. The antiderivative of \( e^x \) is \( e^x \), and the antiderivative of \( e^{-2x} \) is \( -\frac{1}{2}e^{-2x} \), but ensure to account for constants and coefficients.

Step 2: Integrate the first term

The first term is \( 2e^x \). The integral of \( e^x \) is \( e^x \), so the integral of \( 2e^x \) is \( 2e^x \).

Step 3: Integrate the second term

The second term is \( -3e^{-2x} \). The integral of \( e^{-2x} \) is \( -\frac{1}{2}e^{-2x} \), so multiplying by \(-3\), we get \( 3 \times \frac{1}{2}e^{-2x} = \frac{3}{2}e^{-2x} \).

Step 4: Combine to find the general antiderivative

Combine the results of Steps 2 and 3 to express the general antiderivative:\[ \int (2e^x - 3e^{-2x}) \, dx = 2e^x + \frac{3}{2}e^{-2x} + C \]Here, \( C \) is the constant of integration.

Step 5: Verify by differentiation

Differentiate \( 2e^x + \frac{3}{2}e^{-2x} + C \) with respect to \( x \). Differentiating:- The derivative of \( 2e^x \) is \( 2e^x \).- The derivative of \( \frac{3}{2}e^{-2x} \) is \( \frac{3}{2}\times(-2)e^{-2x} = -3e^{-2x} \).- The derivative of \( C \) is 0.Hence, the derivative\( = 2e^x - 3e^{-2x} \), matching the original function, confirming our solution.

Key Concepts

Antiderivative

When working with an indefinite integral, we're in search of its antiderivative. An antiderivative is a function whose derivative equals the given function. For example, the function that gives us back the original given expression when differentiated. It's like finding the reverse process of differentiation. In simple terms, if differentiation is making a path, finding the antiderivative is retracing your steps.

• The antiderivative of a simple exponential function like \( e^x \) is \( e^x \), since differentiating \( e^x \) gives \( e^x \) back.
• When integrating more complex expressions like \( 2e^x \), the process involves integrating \( e^x \) and then multiplying by the constant, resulting in \( 2e^x \).
• Don’t forget the constant of integration, denoted by \( C \). It accounts for all possible vertical shifts of your antiderivative function on a graph.

Finding antiderivatives involves practicing recognizing the patterns of common functions, which builds intuition over time.

Differentiation

Differentiation is the process of finding the derivative of a function. It tells us how a function changes as its input changes. In simpler terms, it gives us the rate of change or the slope of a function at any given point.
Differentiation serves as a tool to confirm if our antiderivative is correct. By differentiating the antiderivative, we should return to the original function.
Consider the derivative steps applied to the antiderivative found in the solution:

• The derivative of \( 2e^x \) with respect to \( x \) is \( 2e^x \). That's because \( e^x \) remains the same when differentiated.
• The derivative of \( \frac{3}{2}e^{-2x} \) includes using the chain rule. Differentiate \( e^{-2x} \) to get \(-2e^{-2x}\), and multiply by \( \frac{3}{2} \) to get \(-3e^{-2x}\).
• Don’t forget: the derivative of a constant is zero, which is why the \( C \) vanishes in differentiation.

Using differentiation, verifying our solution ensures both precision and understanding of the integral we computed.

Exponential Function

The exponential function is a crucial building block in mathematics, especially within calculus. It's characterized by its base, which often is the irrational number \( e \), approximately equal to 2.71828.
Exponential functions have unique properties:

• They grow fast! Any positive value raised to a variable power grows exponentially.
• The derivative of \( e^x \) is \( e^x \), making it unique since it's its own derivative and antiderivative.
• Exponential functions are everywhere—modeling populations, radioactive decay, or finance calculations, being highly applicable in real-world situations.

In the context of solving integrals, knowing how they integrate and differentiate is essential. For instance, integrating \( e^x \) is straightforward, but accounts for expressions like \( e^{-2x} \), the exponent impacts both the antiderivative and its differentiation.
Use these key insights to tackle problems involving exponential functions, making sure to handle the operations confidently and accurately.