Problem 68 - Calculus Early Transcendentals

Question

Use a change of variables to evaluate the following integrals. $$\int \frac{e^{2 x}}{e^{2 x}+1} d x$$

Step-by-Step Solution

Answer

Answer: Use the substitution method with $u = e^{2x} + 1$. The solution to the integral is $\frac{1}{2} \ln{|e^{2x} + 1|} + C$.

Step 1: Choose an appropriate substitution

Let's let a new variable u be: $$u = e^{2x} + 1$$

Step 2: Differentiate the substitution with respect to x

Differentiate u with respect to x: $$\frac{du}{dx} = 2 e^{2x}$$ Rearrange this equation to get dx: $$dx = \frac{1}{2e^{2x}}du$$

Step 3: Substitute the new variable into the integral

Now, we will replace the variables in the integral with our new variable u and du: $$\int \frac{e^{2 x}}{u} \frac{1}{2e^{2x}} du$$ Simplify the integral expression: $$\frac{1}{2} \int \frac{1}{u} du$$

Step 4: Evaluate the integral

This integral is now a basic integration problem: $$\frac{1}{2} \int \frac{1}{u} du = \frac{1}{2} \ln{|u|} + C$$

Step 5: Substitute the original variables back in

Replace u with the original expression in terms of x: $$\frac{1}{2} \ln{|e^{2x} + 1|} + C$$ This is the final answer for the integral.

Key Concepts

change of variables

The change of variables is a pivotal concept in calculus, used to simplify complex integrals through substitution. When directly integrating a function seems challenging, we can introduce a new variable to rewrite the integral in a simpler form.

• In the case of our original function, it's given as \( \int \frac{e^{2x}}{e^{2x}+1} \, dx \). Here, a direct approach to solving seems difficult due to the complexity in the exponent.
• To apply change of variables or substitution, select a part of the integral expression to act as your new variable \(u\). This choice often stems from identifying a nested function or composite part of the given function. Here, choosing \( u = e^{2x} + 1 \) streamlines the integration process.
• It requires differentiating this \( u\) with respect to \( x\) and finding \(dx\) in terms of \(du\). In this case, we get \( \frac{du}{dx} = 2e^{2x} \) which simplifies to \( dx = \frac{1}{2e^{2x}} du \).

The new variable \( u \) should encompass much of the complexity, making it easier to perform the integration and substitute back once evaluated.

definite integrals

Definite integrals refer to the evaluation of the integral of a function over a specified interval. Unlike indefinite integrals, they yield a real number that represents the net area under the curve of a function.

• In practice, after performing a suitable substitution or integration directly, definite integrals require you to use the limits of integration, applying them to the antiderivative.
• The outcome is a numerical value that may represent a significant physical quantity such as area, volume, or even total change in certain applications.
• When using a change of variables approach, it's vital to adjust the limits from the original ones based on \( x \) to the transformed limits in terms of \( u \), using the substitution. This ensures the integration process remains valid and accurate.

While this solution involved an indefinite integral, understanding definite integrals helps in integrating real-world applications, where precise numerical evaluations are necessary.

indefinite integrals

Indefinite integrals find the antiderivative of a function, returning a general formula that includes a constant of integration, \(C\). The outcome is not a specific value but a family of functions.

• For the integral \( \frac{1}{2} \int \frac{1}{u} \, du \), it yields an antiderivative of the form \( \frac{1}{2} \ln{|u|} + C \).
• This solution inherently includes \( + C\) to account for the indefinite nature of antiderivatives since differentiating it must yield the original function through all potential vertical shifts of the function graph.
• In practical terms, the expression \( \frac{1}{2} \ln{|e^{2x} + 1|} + C \) is the general solution before applying any boundary conditions or added specificity that might arise in definite contexts.

The concept of indefinite integrals is crucial in mathematics, enabling the solution of integration problems with various potential outcomes depending on specific conditions or additional solved information.