Problem 53 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the definite integral by hand. Then use a symbolic integration utility to evaluate the definite integral. Briefly explain any differences in your results. $$ \begin{aligned} &\int_{0}^{3} \frac{2 e^{x}}{2+e^{x}} d x\\\ &\text { Exercises } 55-60, \text { eva } \end{aligned} $$

Step-by-Step Solution

Answer

Evaluating the integral gives \( 2*ln\left(\frac{e^3 + 2}{2}\right) \). The result obtained manually and from the symbolic integration tool should be the same, unless the tool simplifies the function in a different manner.

Step 1: Start Integrating

Begin by applying the technique of integration by substitution. Let \( u = 2 + e^x \). Derive \( du/dx \) from this, which gives \( du/dx = e^x \). Then solve for \( dx \), we get \( dx = du/e^x \).

Step 2: Substitute Variables

Substitute \( u \) and \( dx \) in the integral to obtain \( \int_{2}^{e^3 + 2} \frac{2 du}{u} \). This modifies the integral into a simpler form, which is easier to solve.

Step 3: Perform the Integration

The integral now appears as the natural logarithm, ln(u). And the integration is performed over the new limits 2 to \( e^3 + 2 \), which gives \( 2[ln(e^3 + 2) - ln(2)] \).

Step 4: Simplify the Integral

Simplify the expression to \( 2*ln\left(\frac{e^3 + 2}{2}\right) \).

Step 5: Evaluate using Symbolic Integration Utility

Now we can verify the result using a symbolic integration utility (like Mathematica, Symbolab or an online integral calculator). The result should be the same unless the tool simplifies the function differently, but ultimately they should represent the same value.

Key Concepts

Integration by Substitution

Integration by substitution is a method used to solve integrals more easily by transforming a complicated expression into a simpler one. This technique resembles the chain rule in differentiation, where you identify a part of the integrand that can be substituted with a single variable, usually creating a cleaner equation.
For instance, in the exercise given, we have the integral of the form \(\int \frac{2 e^{x}}{2+e^{x}} dx\). To simplify it, we let \( u = 2 + e^x \).
This substitution transforms our integral into \(\int \frac{2 du}{u}\), a more manageable form.

• First, choose a substitution \(u\) that simplifies the integral.
• Compute the derivative \(du/dx\) and rearrange to find \(dx\).
• Replace original variable terms in the integral with \(u\) terms.

This approach bypasses complex expressions and gets us closer to finding the antiderivative, easing the integration process.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a logarithmic function with base \(e\), where \(e\) is the irrational number approximately equal to 2.718.In calculus, this function often appears in integration, especially when dealing with exponential functions.
Once the integral is rewritten using substitution, it transforms into a straightforward natural logarithm form, \(\int \frac{2 du}{u}\).
This type of integral evaluates to \(2 \ln(u)\), since the integral of \(\frac{1}{u}\) is \(\ln(|u|)\).

• It represents the area under the curve of \(\frac{1}{x}\) between positive real numbers.
• Useful in simplifying expressions involving exponential growth/decay.

The natural logarithm is essential when solving integrals that involve division by a variable, simplifying the process by handling it as a straightforward result.

Symbolic Integration

Symbolic integration involves finding the exact value of an integral using symbols, rather than numerical approximations.This contrasts with numerical integration, which estimates the value using numbers and limits its precision.
Symbolic tools apply rules of calculus to find antiderivatives expressed in variables and constants.These tools include software like Mathematica or online calculators like Symbolab.
After performing an integration by substitution in our problem, a symbolic integration utility checks the work by evaluating \(\int \frac{2 e^{x}}{2+e^{x}} dx\) within the specified limits \([0,3]\).

• Allows comparison to ensure manual solutions align with numerical tools.
• Helps catch any algebraic errors and simplifies complex symbolic expressions.

Although symbolic integration may appear different due to simplifications, the results fundamentally convey the same mathematical value.

Integration Techniques

There are several techniques in integration that are employed based on the nature of the function being integrated.These methods help break down complex integrals into simpler forms.
Besides integration by substitution, other techniques include:

• Integration by Parts: Useful for products of functions, formulated from the product rule in differentiation.
• Partial Fraction Decomposition: Decomposes rational functions into simpler fractions.
• Trigonometric Substitution: Useful for integrals involving \(\sqrt{a^2 - x^2}\), \(\sqrt{a^2 + x^2}\), and \(\sqrt{x^2 - a^2}\).
• Special Integrals: Certain functions have standard integrals that can be directly applied.

Choosing the right technique streamlines the process and ensures accurate results.In the given exercise, substitution is chosen due to the presence of an exponential function, ideal for relating one component to \(u\). By mastering these techniques, students can tackle a wide array of integrals with confidence.