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Chapter 5: Problem 90

Find the indefinite integral.\(\int \frac{e^{2 x}}{1+e^{2 x}} d x\)

Short Answer

Expert verified
The result of the integral is \(1/2 \ln |1+ e^{2x}|\)

Step by step solution

01

Substitution

Set \(u = 1 + e^{2x}\). The derivative of \(u\) with respect to \(x\) is \(du/dx = 2e^{2x}\). Thus, \(1/2 du = e^{2x} dx\). Substituting gives, \(\int \frac{1/2 du}{u}\).
02

Integration

Now that the integral has been simplified through the substitution, we can integrate it directly. The integral of \(1/u\) with respect to \(u\) is \(\ln |u|\), thus our integral becomes \(1/2 \int du/u = 1/2 \ln |u|\).
03

Back substituting

After completing the integration, substitute back to remove the \(u\) introduced in step 1 with the original variable \(x\). Therefore, the solution to the integral is \(1/2 \ln |1+ e^{2x}|\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Indefinite Integrals
Indefinite integrals are a fundamental concept in calculus, representing the process of finding an antiderivative of a function. An antiderivative is essentially the opposite of a derivative and helps in determining a function given its derivative.
Key points to remember about indefinite integrals include:
  • The result includes a constant of integration, denoted as "C", since integrating removes information about the constant that might have been lost during differentiation.
  • Indefinite integrals are expressed using the integral sign \ ( \int \cdot \,dx \ ), indicating the function to be integrated with respect to \(x\).
  • The process is also referred to as 'antidifferentiation.'
In our exercise, we are asked to evaluate \( \int \frac{e^{2x}}{1+e^{2x}} \, dx \). To tackle this, we need to simplify the expression before finding its antiderivative.
Substitution Method
The substitution method, also known as "u-substitution," is a technique used to simplify complex integrals. It involves replacing a part of the integral with a new variable to make the expression easier to integrate.
Here's how substitution works:
  • Identify a part of the integral that can be replaced with a new variable \(u\).
  • Calculate \(du\), the differential of \(u\), in terms of \(dx\). This often simplifies the integral.
  • Rewrite the integral in terms of \(u\) and \(du\).
  • Once the new expression is simpler, integrate with respect to \(u\).
In our exercise, we set \( u = 1 + e^{2x} \), leading to \( du = 2e^{2x} dx \) and consequently \( \frac{1}{2} du = e^{2x} \, dx \). By substituting, the integral transforms to \( \int \frac{1}{2} \, du/u \), making it manageable.
Natural Logarithm
Natural logarithms, denoted \( \ln \), are logarithms with base \(e\), where \(e\) is approximately 2.71828. They play a crucial role in calculus, particularly in solving integrals.Considerations for natural logarithms include:
  • The natural logarithm of a positive number \(u\) is represented as \(\ln |u|\), which helps manage integrals involving \(1/u\).
  • When integrating functions of the form \(1/u\), the result is \(\ln |u| + C\), where \(C\) is the integration constant.
  • Natural logarithms have properties similar to other logarithms, such as \(\ln(ab) = \ln a + \ln b\), providing flexibility in algebraic manipulations.
In our exercise, after substitution, we integrate \( \int 1/u \, du \) to obtain \( \ln |u| + C \). Finally, substituting back for \(u\) gives us \( \frac{1}{2} \ln |1+e^{2x}| + C\) as the final answer.

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