Problem 65 - Applied Calculus

Question

Find the indefinite integrals. $$\int\left(e^{x}+5\right) d x$$

Step-by-Step Solution

Answer

The indefinite integral is \( e^x + 5x + C \).

Step 1: Understand the Integral

The integral you're tasked with finding is \( \int (e^x + 5) \, dx \). This is an indefinite integral where you need to determine a function whose derivative gives you the original expression \( e^x + 5 \).

Step 2: Integrate Each Term Separately

Since the integral is a sum of terms \( e^x \) and \( 5 \), you can integrate each term separately: \( \int e^x \, dx \) and \( \int 5 \, dx \).

Step 3: Integrate \( e^x \)

The integral of \( e^x \) with respect to \( x \) is simply \( e^x \), because the derivative of \( e^x \) is \( e^x \). This gives us the result \( e^x \).

Step 4: Integrate the Constant 5

For the constant \( 5 \), the integral is \( \int 5 \, dx = 5x \), because the derivative of \( 5x \) is \( 5 \).

Step 5: Combine Results

Add the results from Step 3 and Step 4 together; thus, the indefinite integral of \( e^x + 5 \) is \( e^x + 5x + C \), where \( C \) is the constant of integration.

Key Concepts

Understanding Integration Techniques

Integration is the reverse process of differentiation. It involves finding a function whose derivative is the given expression. This process can be tricky, as there are several techniques to consider. One of the simplest is the **term-wise integration**.

In our example, the integral of the expression \( e^x + 5 \) is split into the sum of two simpler integrals, \( \int e^x \, dx \) and \( \int 5 \, dx \). By integrating each term separately, you can solve the integral more easily.

• The integral of \( e^x \) is itself, which is a straightforward case due to the nature of exponential functions.
• The integral of a constant \( 5 \) is more approachable by recognizing that it can be rewritten as \( 5 \times 1 \), leading to the integral form \( 5x \).

Remember that these techniques are powerful tools to break complex integrals into parts you can handle one at a time.

Exploring Exponential Functions

Exponential functions like \( e^x \) are unique because their rate of change is proportional to their value. The function \( e^x \) has some distinct properties that make it stand out in calculus.

• The derivative of \( e^x \) is \( e^x \), which means the slope of the curve is the same no matter the value of \( x \).
• This property makes exponential functions easy to work with when integrating, as the antiderivative is also \( e^x \).

These functions are important in modeling real-world scenarios where growth or decay rate is proportional to the current value, such as in population growth or radioactive decay.

Role of Constants of Integration

When solving indefinite integrals, you will often see a "+ C" at the end of your solution. This symbolizes the constant of integration, crucial for indefinite integrals.

During differentiation, any constant added to your function will disappear, as the derivative of a constant is zero. Hence, when you're performing integration, you're retrieving the most general form of the original function, which could have had any constant added to it.

• The \(+ C\) ensures that all possible originals are accounted for.
• Without this constant, your solution only represents one specific antiderivative rather than the infinite family of them.

Always remember to include the "+ C". Without it, your integral solution remains incomplete, omitting the breadth of possibilities it should cover.