Problem 29 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int \frac{e^{x} d x}{4+e^{x}} $$

Step-by-Step Solution

Answer

\( \int \frac{e^x}{4+e^x} \, dx = \ln |4 + e^x| + C \)

Step 1: Identify the Integration Method

The given integral is \( \int \frac{e^x}{4+e^x} \, dx \). Notice the form of the integrand suggests that substitution might be a useful strategy. Specifically, substituting the denominator might help simplify the expression.

Step 2: Choose a Substitution Variable

Let \( u = 4 + e^x \). Then, the derivative \( du = e^x \, dx \). This substitution simplifies the differential \( dx \) part of the integral when expressed in terms of \( u \).

Step 3: Rewrite the Integral in Terms of \( u \)

Substitute \( u = 4 + e^x \) and \( du = e^x \, dx \) into the integral. The integral becomes: \( \int \frac{1}{u} \, du \).

Step 4: Integrate the Simplified Expression

The integral \( \int \frac{1}{u} \, du \) is a standard form and is equal to \( \ln |u| + C \), where \( C \) is the integration constant.

Step 5: Substitute Back to Original Variable

Replace \( u \) with the original expression \( 4 + e^x \) to return to the variable \( x \). So, the integral becomes \( \ln |4 + e^x| + C \).

Step 6: Verify by Differentiation

Differentiate the result from Step 5 with respect to \( x \). The derivative of \( \ln |4 + e^x| \) is \( \frac{d}{dx}[\ln(4 + e^x)] \). Applying chain rule, the derivative is \( \frac{e^x}{4 + e^x} \), which is the original integrand, confirming the solution is correct.

Key Concepts

Definite and Indefinite Integrals

In calculus, integration is the process of finding an integral of a function, which can be thought of as the "reverse" of differentiation. Integrals come in two main forms: definite and indefinite.

• Indefinite Integrals: These do not have specified upper and lower limits of integration. They represent a family of functions and typically include a constant of integration, denoted as "C". The example provided, \( \int \frac{e^x}{4+e^x} \, dx \), is such an indefinite integral, as we integrate without boundaries.
• Definite Integrals: Unlike indefinite integrals, definite integrals have upper and lower limits. They are used to calculate the area under a curve within a specified interval and result in a specific numeric value. They do not include a constant of integration.

Understanding the difference between these two types is crucial as it affects how problems are approached and solved.
The outcome of evaluating an indefinite integral, like in our problem, is a formula that represents the antiderivative of the function. This formula can then be evaluated over specific intervals for definite evaluations if needed.

Chain Rule

The chain rule is an essential tool in calculus for finding derivatives of composite functions. It's particularly useful when dealing with functions written in terms of other functions, as is often the case in substitution methods.
In the exercise given, we used substitution to change variables from \( x \) to \( u = 4 + e^x \). To differentiate after integrating, the chain rule helps us link these variables back and verify correctness.

• The chain rule formula: If we have a composite function \( f(g(x)) \), then the derivative is \( f'(g(x)) \cdot g'(x) \).
• For our solution, after substituting and integrating, we reach \( \ln|4 + e^x| \). By the chain rule, differentiating this yields: \( \frac{1}{4+e^x} \cdot e^x = \frac{e^x}{4+e^x} \).

This simplification demonstrates how versatile the chain rule is in connecting back to the function's original form, offering a reliable method for verifying integrations and substitutions performed.

Differentiation Verification

Checking a computed integral by differentiation ensures the correctness of the solution. It involves taking the derivative of the result to see if it matches the original integrand.
When we complete an indefinite integral like \( \int \frac{e^x}{4+e^x} \, dx = \ln|4+e^x| + C \), the next logical step is to differentiate \( \ln|4+e^x| \):

• According to differentiation rules and applying the chain rule, the derivative becomes \( \frac{1}{4+e^x} \cdot e^x = \frac{e^x}{4+e^x} \).
• This result matches the original function \( \frac{e^x}{4+e^x} \), verifying that the integration was executed correctly.

Verification via differentiation not only offers reassurance but also deepens one's understanding of differential calculus concepts. It bridges integration and differentiation, showcasing their interrelated nature and cementing the accuracy of solution methods used.