Problem 21 - Calculus: Early Transcendental Functions

Question

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

Step-by-Step Solution

Answer

The result of the integration is \(e^{2x} - \ln |e^{2x} + 1| + C\).

Step 1: Identify the Inner Function and Set Up the Substitution

Recognize the complex part of the function. In the given integral, \(e^{2x}\) is the more complex part. Set \(u = e^{2x}\). Then calculate the derivative of \(u\), \(du = 2e^{2x}dx\). Now the original integral in \(du\) form is twice of \(\int \frac{u}{1+u} du\).

Step 2: Handling the Integral

Split the fraction under the integral into two simpler parts by using the formula \(1+u = u + 1\), giving \(\int \frac{u}{u + 1} du = \int \frac{u + 1 - 1}{u + 1} du\). This simplifies to \(\int du - \int \frac{1}{u + 1} du = u - \ln |u + 1| + C\).

Step 3: Substituting Back

Substitute back the original variable \(x\) using the \(u\)-substitution. Instead of \(u\), insert \(e^{2x}\), to obtain the final answer as \(e^{2x} - \ln |e^{2x} + 1| + C\).

Key Concepts

Substitution Method

In integral calculus, the substitution method is a powerful technique that simplifies the process of solving integrals by reworking them into a more manageable form. The essence of this method lies in identifying a part of the integral that can be substituted with a new variable, often denoted as \( u \). Substitution is particularly useful when there is a composite function, ensuring the integral transforms into a simpler expression.

When using substitution:

• Identify a part of the integrand as \( u \), focusing on functions where a derivative matches another part of the integrand.
• Calculate the derivative of \( u \) with respect to \( x \), to find \( du \).
• Rewrite the integral in terms of \( u \) and \( du \), often simplifying the process.

In our exercise, recognizing \( e^{2x} \) as \( u \) simplifies the original integral. This leads to the equation \( u = e^{2x} \), with the derivative \( du = 2e^{2x} dx \). The integral transforms into a simpler form, making it easier to solve.

Integral Calculus

Integral calculus revolves around the concept of integration, which is essentially the inverse operation of differentiation. The goal of integration is often to find the area under a curve or to calculate accumulated quantities.

Two main types of integration are:

• Indefinite Integrals: These do not have specified limits and include a constant of integration \( C \).
• Definite Integrals: These have limits and provide a numeric result, representing areas or accumulated values.

In the context of our exercise, we are dealing with an indefinite integral \( \int \frac{e^{2x}}{1+e^{2x}} dx \). By using substitution, the integral is expressed in terms of \( u \), transforming it into a simpler problem to solve using fundamental integral calculus techniques.

Logarithmic Integration

Logarithmic integration is a technique often employed to solve integrals involving fractions where the numerator is the derivative of the denominator. This can simplify the integration process, especially when dealing with expressions of the form \( \frac{1}{x} \), yielding a logarithmic function as part of the solution.

For example, the integral of \( \frac{1}{u+1} \) is \( \ln |u+1| \). In our exercise, after performing the substitution and simplifying the expression, we encounter this logarithmic form. The original expression splits into terms manageable by basic integration rules.

• Separate the terms: \( \int \frac{u + 1 - 1}{u + 1} du \).
• Simplify, leading to an integration that includes a logarithm: \( \int du - \int \frac{1}{u + 1} du \).

Ultimately, this method provides solutions involving natural logarithms, rendering seemingly complex integrals more straightforward.