Problem 29 - Thomas Calculus in SI Units

Question

Evaluate the integrals in Exercises \(29-50.\) $$\int\left(e^{3 x}+5 e^{-x}\right) d x$$

Step-by-Step Solution

Answer

\( \int (e^{3x} + 5e^{-x}) \, dx = \frac{1}{3}e^{3x} - 5e^{-x} + C \)

Step 1: Identify and Separate the Terms

The integral is given by \( \int (e^{3x} + 5e^{-x}) \, dx \). This can be separated into two integrals: \( \int e^{3x} \, dx \) and \( 5 \int e^{-x} \, dx \). This approach simplifies solving the integrals separately.

Step 2: Integrate the First Term

Consider the first term \( \int e^{3x} \, dx \). To integrate \( e^{3x} \), use the formula for integrating exponential functions: \( \int e^{ax} \, dx = \frac{1}{a}e^{ax} + C \). Therefore, for our term \( \int e^{3x} \, dx = \frac{1}{3} e^{3x} + C_1 \).

Step 3: Integrate the Second Term

Next, consider \( 5 \int e^{-x} \, dx \). Again, use the exponential integration formula. Here, \( a = -1 \), so \( \int e^{-x} \, dx = -e^{-x} + C_2 \). Therefore, \( 5 \int e^{-x} \, dx = -5e^{-x} + C_3 \).

Step 4: Combine the Results

Add the results from Steps 2 and 3. Combine constants into one constant, since the sum of arbitrary constants is still an arbitrary constant. Therefore, the combined result is: \( \int (e^{3x} + 5e^{-x}) \, dx = \frac{1}{3} e^{3x} - 5e^{-x} + C \).

Key Concepts

Exponential Functions

Understanding exponential functions is crucial when dealing with integrals involving terms like \( e^{3x} \) and \( 5e^{-x} \). An exponential function is any mathematical expression in the form \( e^{ax} \), where \( e \) represents Euler's number (approximately 2.718), and \( a \) is a constant determining the rate of growth or decay.

• The base \( e \) is a constant that appears frequently in mathematics due to its unique characteristics in the context of growth and decay processes.
• The exponent, such as \( 3x \) or \(-x\), represents the variable part of the function, which impacts the behavior of the function over different values of \( x \).

When \( a \) is positive (e.g., \( e^{3x} \)), the function models exponential growth. When \( a \) is negative (e.g., \( 5e^{-x} \)), it shows exponential decay. Recognizing these patterns is key to solving integrals involving exponential terms.

Integral Evaluation

Evaluating integrals with exponential functions often involves applying a standard formula. For any exponential function \( e^{ax} \), the integral is calculated using the formula:\[\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\]
Here's a breakdown of the steps:

• Identify the constant \( a \) in your exponential term \( e^{ax} \).
• Apply the integral formula directly to get the antiderivative.
• Multiply the result by the constant outside the integral, when applicable, as done with \( 5 \int e^{-x} \, dx \).

This process simplifies the complexity typically encountered with exponential integrals. Always ensure to apply the right formula accurately to each term.

Constants of Integration

In the process of integration, we often end a solution with a constant past the integral sign, denoted \( C \) or other symbols like \( C_1, C_2, \) etc.

• These constants arise because integration introduces an arbitrary constant that reflects the family of antiderivatives.
• During integration, the constant \( C \) compensates for the unknown initial value lost when differentiating, providing flexibility to fit initial conditions if specified.
• Once you integrate different terms, combine all constants into a single constant \( C \), as they are indeterminate.

Understanding the role of constants of integration ensures that the solution accounts for every possible scenario of the original differential equation.