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Chapter 6: Problem 3

Calculate the integral, if it converges. If it diverges, enter diverges for your answer. \(\int_{-\infty}^{1} \frac{e^{5 x}}{1+e^{5 x}} d x=\) ___________.

Short Answer

Expert verified
\frac{1}{5} \ln(1+e^5)

Step by step solution

01

Understand the Problem

We need to evaluate the improper integral \(\[ \int_{-\text{infty}}^{1} \frac{e^{5x}}{1+e^{5x}} \, dx \]\ \). Since the lower limit is \(-\text{infty}\), we need to determine if the integral converges or diverges.
02

Change of Variable

Let's make a substitution to simplify the integral. Let \( u = e^{5x} \). Then, \( du = 5e^{5x} dx \), and \[ dx = \frac{du}{5u} \].
03

Adjust the Limits

When \( x \) approaches \(-\text{infty} \), \( u \) approaches 0. When \( x = 1 \), \( u = e^5 \). So the new limits are from 0 to \ e^5 \.
04

Substitute and Simplify

Substituting the variables, the integral becomes: \[ \int_{0}^{e^5} \frac{u}{1+u} \cdot \frac{1}{5u} \, du \] = \ \frac{1}{5} \int_{0}^{e^5} \frac{1}{1+u} \, du \.
05

Integrate

Integrate the simplified form: \[ \frac{1}{5} \int_{0}^{e^5} \frac{1}{1+u} \, du = \frac{1}{5} \left[ \ln|1+u| \right]_{0}^{e^5} \]
06

Evaluate the Definite Integral

Evaluate the integral at the limits: \[ \frac{1}{5} \left( \ln(1+e^5) - \ln(1+0) \right) = \frac{1}{5} \left( \ln(1+e^5) - \ln(1) \right) = \frac{1}{5} \ln(1+e^5) \].
07

Conclusion

Since \( \frac{1}{5} \ln(1+e^5) \) is a finite value, the integral converges.

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Key Concepts

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

Integration Techniques
When dealing with improper integrals, choosing the right integration technique can make the problem more manageable. In our example, we used a substitution to simplify the integral. Substituting variables is often helpful when the integrand contains composite functions or when the limits are infinite. The substitution helps transform the integral into a simpler form, as we did by letting \(u = e^{5x}\). This allowed us to change the variable of integration and adjust the limits accordingly. Understanding and applying the appropriate technique is crucial for solving complex integrals efficiently.
Change of Variables
Changing variables is a powerful method used to simplify integrals. In our exercise, we used the substitution \(u = e^{5x}\) to transform the integral into a form that's easier to handle. By doing this, the differential \(dx\) was converted into \(\frac{du}{5u}\). The limits of integration also changed from \(x\) to \(u\). This step is essential in linking the original integral's boundaries to the new variable. Changing variables helps us to straighten out complex expressions and often reduces the difficulty of evaluating the integral.
Convergence and Divergence
When dealing with improper integrals, determining whether an integral converges or diverges is essential. Convergence means that the integral approaches a finite value, while divergence means it does not. In our exercise, after performing the substitution and integration, we evaluated the definite integral from 0 to \(e^5\). The expression \(\frac{1}{5} \ln(1+e^5)\) is finite, indicating that the integral converges. It's essential to check the limits of the integral and ensure that the result is finite to confirm convergence. If the result is infinite or undefined, the integral diverges.

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Most popular questions from this chapter

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