Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int\left(e^{2 x}+e^{-3 x}\right) d x$$

Step-by-Step Solution

Answer

The indefinite integral of the given expression is: \[\frac{1}{2} e^{2x} - \frac{1}{3} e^{-3x} + C\]

Step 1: Separate the terms for integration

We will integrate each term in the given expression separately: \[\int\left(e^{2x} + e^{-3x}\right) dx = \int e^{2x} dx + \int e^{-3x} dx\]

Step 2: Integrate the first term

To integrate the first term, apply the formula for the integration of an exponential function: \[\int e^{2x} dx = \frac{1}{2} e^{2x} + C_1\]

Step 3: Integrate the second term

Now, apply the formula for the integration of an exponential function to the second term: \[\int e^{-3x} dx = -\frac{1}{3} e^{-3x} + C_2\]

Step 4: Combine the results and write the final answer

Combine the results from steps 2 and 3, and remember to add a constant of integration to represent the family of antiderivatives: \[\int\left(e^{2x} + e^{-3x}\right) dx = \frac{1}{2} e^{2x} - \frac{1}{3} e^{-3x} + C\] The indefinite integral of the given expression is: \[\frac{1}{2} e^{2x} - \frac{1}{3} e^{-3x} + C\]

Key Concepts

Integration of Exponential Functions

In calculus, the integration of exponential functions is a fundamental concept that involves finding the antiderivative or integral of exponential expressions. An exponential function can generally be defined as \[e^{ax}\] where \(a\) is a constant.
When integrating such functions, the formula used is:

• \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \)

Here, \(C\) stands for the constant of integration which is an essential part of indefinite integrals. It's important to note:

• The exponent's coefficient \(a\) determines the scaling factor in front of the result.
• The integral maintains the same base of \(e\), but the coefficient in the exponent affects the final expression.

For example, in our given problem, \(\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C\) uses this integration formula. Applying this knowledge assists in smoothly handling various exponential functions.

Antiderivatives

Antiderivatives are also referred to as indefinite integrals. They are the 'reverse' operation to derivatives in calculus.
Finding an antiderivative means identifying a function whose derivative returns the original function.

• An antiderivative of \(f(x)\) is a function \(F(x)\) such that \(F'(x) = f(x)\).
• For exponential functions like \(e^{2x}\), the antiderivative is \(\frac{1}{2}e^{2x} + C\).
• The constant \(C\) accounts for the family of functions that share this characteristic, as derivatives of constants are zero.

In our task, recognizing \(e^{2x}\) and \(e^{-3x}\) separately allowed us to find antiderivatives for each part of the expression, ensuring that when these are added, we have the correct indefinite integral.

Calculus Steps in Problem Solving

To tackle calculus problems effectively, it's helpful to follow structured steps ensuring clarity and correctness. Let's outline these using our example:**Step 1: Decompose Complex Expressions**
Initially, separate distinct terms, as we did with \(e^{2x}\) and \(e^{-3x}\). This helps to simplify and individually address each part.**Step 2: Apply Known Integration Formulas**

• Use specific formulas for each term, here applying \(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\).
• Recognize patterns that match known integral solutions, simplifying the process significantly.

**Step 3: Combine and Simplify**
Add the individual results back into a single expression. Ensure to include the constant of integration, \(C\), which represents an entire family of solutions.
In our example, combining \(\frac{1}{2} e^{2x}\) and \(-\frac{1}{3} e^{-3x}\) leads to the final answer: \(\frac{1}{2} e^{2x} - \frac{1}{3} e^{-3x} + C\).By following these methodical steps, solving calculus problems like indefinite integrals becomes more manageable and logical.