Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the integrals. $$ \int \frac{e^{2 x}}{\left(e^{x}+2\right)\left(e^{x}-1\right)^{2}} d x $$

Step-by-Step Solution

Answer

After simplifying the integrand by factoring out \(e^x\) and performing a substitution followed by a partial fraction decomposition and integrating the fractions obtained, the required integral can be separately evaluated and the final solution obtained by substituting back the original variables.

Step 1: Simplify the integrand

This can be done by factoring out \(e^x\) from the integrand which will simplify it. This results in \[ \int \frac{e^{x}}{(e^{x}+2)(1-e^{-x})^{2}} dx \]. Then, substitute \(u = e^{x}\), as \(\frac{du}{dx} = e^{x}\), this turns the \(d x\) into \(du\), and we obtain \[ \int \frac{1}{(u+2)(1-\frac{1}{u})^{2}} du \].

Step 2: Perform partial fraction decomposition

Using the quotient rule of differentiation we perform partial fraction decomposition to further simplify the function. We would break the function into simpler fractions. We look for two functions \(A\) and \(B\) such that \[ \frac{1}{(u+2)(1-\frac{1}{u})^{2}} = \frac{A}{u+2}+ \frac{B}{(1-\frac{1}{u})^{2}} \].

Step 3: Solve for A and B

By equating coefficients, the values of \(A\) and \(B\) will be determined and afterwards the integrals can be evaluated separately.

Step 4: Integrate the fractions

Each of the fractions resulting from the partial fraction decomposition can then be integrated separately.

Step 5: Back substitute the value of \(u\)

Substitute \(u = e^{x}\) back into the integral. The result is the final solution to the integral.

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus, used to find areas, compute volumes, and solve differential equations. Various techniques have been developed to simplify and solve integrals, since not all functions have straightforward antiderivatives.

• Substitution Method: A common technique where a substitution is made to simplify the integral. Essentially, it involves changing the variable of integration, which can transform a complicated integral into a simpler one.
• Partial Fraction Decomposition: Useful when dealing with rational functions. It involves breaking a complex rational expression into simpler parts that can be integrated separately.
• Integration by Parts: Based on the product rule for differentiation, this technique is handy for products of functions.
• Numerical Integration: Methods such as the Trapezoidal Rule or Simpson's Rule approximate the value of an integral.

In this exercise, we primarily use substitution and partial fraction decomposition to tackle a challenging integral.

Partial Fraction Decomposition

Partial fraction decomposition is an essential method for integrating rational functions, where one polynomial is divided by another. This technique simplifies the integration process by expressing the original rational function as a sum of simpler fractions.
Consider a fraction \[\frac{P(x)}{Q(x)}\] where the degree of \(P(x)\) is less than the degree of \(Q(x)\). Partial fraction decomposition attempts to rewrite this as \[\frac{A}{(u+2)} + \frac{B}{(1-\frac{1}{u})^2}\] for simpler integration.

• Step 1: Factor the denominator, if needed.
• Step 2: Break the expression into a sum of simple fractions.
• Step 3: Solve for the constants by equating coefficients or substituting suitable values for the variable.

This process ensures that each part can be integrated independently, often turning an otherwise intractable problem into a approachable one.

Substitution Method

The substitution method, also known as u-substitution, transforms an integral into a simpler form. It is particularly useful when an integral contains a function and its derivative.
The general idea is to replace a complicating part of the integral with a single variable. In our example, the substitution driven rearrangement simplifies the process:

• Step 1: Identify a substitution that simplifies the integrand; for instance, let \(u = e^x\).
• Step 2: Differentiate your substitution to express dx in terms of du; i.e., \(\frac{du}{dx} = e^x\), so \(dx = \frac{du}{e^x}\).
• Step 3: Rewrite the integral in terms of \(u\) and \(du\).

Once the integration is performed in terms of \(u\), substitute back in terms of \(x\) to complete the process. This technique simplifies many integrals significantly by transforming variables.

Definite and Indefinite Integrals

Integration is used to find both definite and indefinite integrals. Each serves a different purpose.

• Indefinite Integrals: Represent a family of functions with an arbitrary constant \(C\), symbolizing all possible antiderivatives of a function. For example, the indefinite integral \(\int f(x)\,dx\) represents the antiderivative of \(f(x)\).
• Definite Integrals: Calculated between specific bounds and result in a number, representing the net area under a curve from one point to another. They are denoted as \(\int_{a}^{b} f(x)\,dx\).

In this exercise, we are primarily dealing with an indefinite integral, as we are not given limits of integration. After simplifying and integrating the expression, results generally come with an added constant \(C\), representing the family of antiderivatives.