Problem 22 - Calculus: Early Transcendental Functions

Question

Evaluate the integral. $$\int \frac{e^{x}}{1+e^{2 x}} d x$$

Step-by-Step Solution

Answer

The solution of the integral \( \int \frac{e^{x}}{1+e^{2 x}} d x \) is \( x - ln(e^{x}+1) + C \)

Step 1: Identify a Suitable Substitution

In the given integral \( \int \frac{e^{x}}{1+e^{2 x}} d x \), observe that the denominator is slightly bigger than the numerator. Looking closer, the derivative of \( e^{2x} \) is \( 2e^{x} \), which is very close to the numerator. So make a substitution \( u = e^{x} \). This means \( x = ln(u) \) and \( dx = \frac{1}{u} du \). After substitution, the integral becomes \( \int \frac{1}{u+u^2} du \)

Step 2: Double Fraction Decomposition

Split the fraction into partial fractions, \( \int \frac{1}{u+u^2} du = \int \frac{1}{u} - \frac{1}{u+1} du \)

Step 3: Perform the Integral

Now, the integral can be performed more easily with partial fractions, \( \int \frac{1}{u} - \frac{1}{u+1} du = ln|u| - ln|u+1| + C \)

Step 4: Substitute Back the Original Variable

Substituting back the original variable \( u = e^{x} \) leads to the final answer, \( ln|e^{x}| - ln|e^{x}+1| + C = x - ln(e^{x}+1) + C \)

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus, involving the process of finding the integral or antiderivative of a function.
This process is essential in determining the area under a curve, solving differential equations, and in many other applications across physics, engineering, and economics.
There are several techniques of integration which are used depending on the type of function involved.
Some common techniques include the power rule, integration by parts, trigonometric integration, and the method of partial fractions. Each technique has specific scenarios in which it is most effective, and recognizing which method to apply is a critical skill in calculus.

Understanding Different Techniques

Students should be familiar with a variety of integration techniques to handle different integrals effectively. For example, a straightforward polynomial might only need the power rule, while a more complicated rational function might require partial fraction decomposition or trigonometric substitution.

U-Substitution

U-substitution is a technique similar to the reverse chain rule in differentiation and is particularly useful when dealing with composite functions.

Implementing U-Substitution

The main idea behind u-substitution is to simplify an integral by substituting a part of the integrand with a new variable, often denoted as 'u'. By differentiating 'u' with respect to 'x', we can express 'dx' in terms of 'du'. This conversion often simplifies the integral into a more manageable form.
Mastery of u-substitution requires the ability to identify an appropriate substitution that will simplify the integral, as well as proficient algebraic manipulation to convert the entire integral in terms of 'u'. After integrating with respect to 'u', the final step involves substituting back the original variables to achieve the final result.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to simplify the integration of rational functions, which are fractions that involve polynomials in both the numerator and the denominator.

The Decomposition Process

The goal is to break down a complex fraction into a sum of simpler fractions, which are easier to integrate. The process involves rewriting the original fraction into one whose denominator factors into linear or quadratic factors, each of which corresponds to a term in the decomposition. From there, constants (or functions, in the case of non-linear factors) are determined to satisfy the equation.
This is particularly useful when dealing with improper fractions, where the degree of the numerator is greater than or equal to that of the denominator. By using partial fraction decomposition, integrals that appear complex can often be reduced to a series of simpler logarithmic or arithmetical integrals.

Exponential Functions

In calculus, exponential functions are widely studied due to their frequent occurrence in various disciplines.

Characteristics of Exponential Functions

An exponential function is of the form \(f(x) = a^x\), where 'a' is a constant base and 'x' is an exponent. These functions are characterized by their rate of growth or decay, which is constant and proportional to the function's current value.
When integrating exponential functions, one often comes across the base \(e\), which is an important mathematical constant approximately equal to 2.71828. The function \(e^x\) is unique because it is its own derivative and antiderivative, making it particularly easy to work with in differential and integral calculus. Recognizing when and how to manipulate exponential functions with various bases is an important aspect of solving integration problems, and it often involves using logarithms and rules of exponents.

• Rate of Growth/Decay: Exponential functions can model processes that grow or decay at a constant relative rate.
• Special Base e: The base \(e\) has properties that simplify differentiation and integration.
• Integration Techniques: Knowing how to handle \(e^x\) in integration can greatly simplify the solution process.