Problem 49 - 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: Break Down the Integral

The integral given is \( \int (e^{x} + 5) \, dx \). Notice that it is a sum inside the integral, which we can split into two separate integrals: \( \int e^{x} \, dx + \int 5 \, dx \).

Step 2: Integrate the First Term

Integrate the first term, \( \int e^{x} \, dx \). The integral of \( e^{x} \) with respect to \( x \) is \( e^{x} \) because the derivative of \( e^{x} \) is itself.

Step 3: Integrate the Second Term

Next, integrate the second term, \( \int 5 \, dx \). This is a constant, so it integrates to \( 5x \), because the derivative of \( 5x \) is \( 5 \).

Step 4: Combine the Results

Combine the results from Steps 2 and 3. The integral \( \int (e^{x} + 5) \, dx \) becomes \( e^{x} + 5x \). Don't forget to add the constant of integration, \( C \), since this is an indefinite integral. Therefore, the final result is \( e^{x} + 5x + C \).

Key Concepts

Understanding Integration Techniques

Integration is like the opposite of differentiation. It is a way of finding a function when its derivative is known. There are various integration techniques that simplify solving problems:

• Substitution: Changing variables to simplify the integral.
• Integration by Parts: Splitting the integral into simpler parts to solve.
• Sum and Difference: Breaking down sums or differences in integrals.

In our original exercise, the focus is on the 'Sum and Difference' technique. Essentially, we deal with broken down parts of the integral separately, making the overall problem simpler. Once integrated, we combine these parts together at the end. This technique turns out especially useful in problems where we have multiple terms inside the integral.

The Sum of Functions Rule in Integration

The Sum of Functions Rule is quite intuitive: the integral of a sum is the sum of the integrals. This rule allows us to handle complex expressions with ease:

• For any two functions, say, \( f(x) \) and \( g(x) \), the integral \( \int [f(x) + g(x)] dx \) can be separated into \( \int f(x) \, dx + \int g(x) \, dx \).
• This step helps us tackle each function individually, often turning a complex problem into straightforward calculations.

In our given problem \( \int (e^{x} + 5) \, dx \), the sum \( e^{x} + 5 \) is divided into \( \int e^{x} \, dx \) and \( \int 5 \, dx \). By treating each term in isolation, we simplify the integration process, arriving at the primary results for each term and eventually adding them together.

The Role of the Constant of Integration

When dealing with indefinite integrals, we must always remember to include the constant of integration, denoted typically as \( C \). This constant plays a critical role for several reasons:

• An indefinite integral represents a family of functions, all differentiated to reach the original function. The constant accounts for any vertical shifts among these functions.
• It embodies the inherent uncertainty when reversing differentiation since many functions can have the same derivative.

In our example, after integrating \( e^{x} \) and \( 5 \) separately, we arrives at \( e^{x} + 5x \). Adding \( C \) gives us the final expression: \( e^{x} + 5x + C \). Always think of \( C \) as a necessary part of the solution to fully express all possible antiderivatives of the original function.