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Magazine Chapter 28: Problem 15
Integrate each of the given functions. $$\int \frac{1+e^{x}}{x+e^{x}} d x$$
Short Answer
Expert verified
The integral is \( \ln |x + e^x| + C \).
Step by step solution
01
Substitution Preparation
Notice that the expression in the integrand, \( \frac{1+e^{x}}{x+e^{x}} \), includes the term \( x + e^x \). To solve this, a substitution where the derivative of the denominator is in the numerator is helpful. Observe that the derivative of \( x + e^x \) is \( 1 + e^x \), which matches the numerator exactly.
02
Set up the Substitution
Let \( u = x + e^x \). Then, differentiate \( u \) with respect to \( x \): \( \frac{du}{dx} = 1 + e^x \). Thus, \( du = (1 + e^x) dx \). This suggests that our integral can be rewritten using this substitution.
03
Substitute and Transform the Integral
Using the substitution, the integral becomes \( \int \frac{1+e^x}{x+e^x} \, dx = \int \frac{1}{u} \, du \) because \( du = (1 + e^x) dx \) replaces \( dx \).
04
Integrate the Simplified Function
The integral \( \int \frac{1}{u} \, du \) is a standard integral that equals \( \ln |u| + C \), where \( C \) is the constant of integration.
05
Back-Substitute to Original Variable
Substitute back the original expression for \( u \). Since \( u = x + e^x \), the solution to the integral becomes \( \ln |x + e^x| + C \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a powerful tool for solving integrals, especially when the integral contains a composite function. In our example, the integrand \( \frac{1+e^{x}}{x+e^{x}} \) suggests an opportunity for substitution because the derivative of the denominator matches the numerator exactly.
Here鈥檚 a step-by-step breakdown of how to use the substitution method in this context:
Here鈥檚 a step-by-step breakdown of how to use the substitution method in this context:
- First, identify a part of the integral where differentiation will simplify the expression. The denominator \( x+e^x \) is a suitable candidate because its derivative, \( 1+e^x \), appears in the numerator.
- Introduce a new variable \( u \), such that \( u = x + e^x \). This choice simplifies the problem since \( \frac{du}{dx} = 1 + e^x \).
- Replace \( dx \) in the integral with \( du/(1 + e^x) \) based on this derivative. With substitution, the integral turns into \( \int \frac{1}{u} \, du \), which is far simpler.
Antiderivatives
Antiderivatives allow us to reverse the process of differentiation. When we integrate a function, we essentially find its antiderivative. In the given problem, after substitution, we are left with the integral \( \int \frac{1}{u} \, du \).
This function is a standard form whose antiderivative is well-known:
This function is a standard form whose antiderivative is well-known:
- The integral of \( \frac{1}{u} \) is \( \ln |u| \). This makes finding the antiderivative straightforward here.
- The constant of integration, \( C \), is added because integration can result in multiple possible functions differing by a constant.
Differential Calculus
Differential calculus is pivotal in the substitution method discussed earlier. It involves finding derivatives of functions, like when we calculated \( \frac{du}{dx} = 1 + e^x \) from \( u = x + e^x \).
Key points to remember about differential calculus in this context:
Key points to remember about differential calculus in this context:
- Differentiation helps in recognizing patterns that can simplify complex integrals by transforming them into standard forms.
- By accurately finding derivatives, we can make informed substitutions that simplify the integration process.
- It helps us understand how small changes in one variable result in changes in another, aiding our understanding of the relationship within the integrated function.
