Problem 17
Question
Evaluate the indefinite integral. \( \displaystyle \int \frac{e^u}{(1 - e^u)^2} \, du \)
Step-by-Step Solution
Verified Answer
The integral is \( \frac{1}{e^u - 1} + C \).
1Step 1: Identify the Integration Technique
To evaluate the indefinite integral \( \int \frac{e^u}{(1 - e^u)^2} \, du \), we should consider a substitution method, as the integrand involves a nested function structure \((1 - e^u)\). This makes it a good candidate for substitution.
2Step 2: Perform Substitution
We set \( v = 1 - e^u \), which implies \( dv = -e^u \, du \). Rearranging gives \( e^u \, du = -dv \). Substituting these into the integral transforms it into: \[ \int \frac{-dv}{v^2}. \]
3Step 3: Simplify and Integrate
The integral \( \int \frac{-dv}{v^2} \) simplifies to \( -\int v^{-2} \, dv \). Integrating yields: \[ \int v^{-2} \, dv = -v^{-1} + C = -\frac{1}{v} + C. \] So, the result is \( \frac{1}{v} + C \), accounting for the negative sign outside the integral.
4Step 4: Reverse the Substitution
Substitute back \( v = 1 - e^u \) into the result: \( -\frac{1}{v} + C \). Therefore, the integral becomes \( -\frac{1}{1 - e^u} + C \).
5Step 5: State the Final Solution
The indefinite integral evaluates to \( \frac{1}{e^u - 1} + C \), where \( C \) is the constant of integration.
Key Concepts
Integration by SubstitutionNested Function IntegrationConstant of Integration
Integration by Substitution
Integration by substitution is a pivotal method when tackling integrals that contain complex or nested functions. Essentially, substitution simplifies the integration process by transforming the original integral into a simpler form. This transformation is achieved by introducing a new variable in place of a portion of the integrand.
To implement this method, you select a suitable substitution that alleviates the complexity of the integrand. In our exercise, this involves choosing a part of the integrand that, when substituted, will simplify the integral. We chose to set \( v = 1 - e^u \). Consequently, the differential \( dv = -e^u \, du \) enables us to express \( e^u \, du \) in terms of \( dv \), transforming the integral into a more manageable form \( \int \frac{-dv}{v^2} \).
This step essentially "unravels" the nested functions, transforming the integral into a format that's easier to solve. Ensuring the right substitution is chosen is the key to successful integration by this method.
To implement this method, you select a suitable substitution that alleviates the complexity of the integrand. In our exercise, this involves choosing a part of the integrand that, when substituted, will simplify the integral. We chose to set \( v = 1 - e^u \). Consequently, the differential \( dv = -e^u \, du \) enables us to express \( e^u \, du \) in terms of \( dv \), transforming the integral into a more manageable form \( \int \frac{-dv}{v^2} \).
This step essentially "unravels" the nested functions, transforming the integral into a format that's easier to solve. Ensuring the right substitution is chosen is the key to successful integration by this method.
Nested Function Integration
Nested functions typically occur when one function is composed of another, making direct integration a challenge. The exercise presents a classic example of nested functions, with the integrand \( \frac{e^u}{(1-e^u)^2} \) consisting of the exponential function \( e^u \) nested within a rational expression.
The complexity of nested functions often necessitates the use of substitution to identify a simpler function structure, enabling straightforward integration. Substitution simplifies these complicated nested expressions by breaking them down into singular components. For instance, replacing \( (1-e^u) \) with a single variable \( v \) straightforwardly reveals the integral's structure, making it manageable. This restructuring is vital as it resolves the intertwining components into a coherent, solvable form.
Understanding this approach to handle nested functions is crucial, as directly attempting to integrate such functions is typically not feasible without simplification.
The complexity of nested functions often necessitates the use of substitution to identify a simpler function structure, enabling straightforward integration. Substitution simplifies these complicated nested expressions by breaking them down into singular components. For instance, replacing \( (1-e^u) \) with a single variable \( v \) straightforwardly reveals the integral's structure, making it manageable. This restructuring is vital as it resolves the intertwining components into a coherent, solvable form.
Understanding this approach to handle nested functions is crucial, as directly attempting to integrate such functions is typically not feasible without simplification.
Constant of Integration
In any indefinite integral, the constant of integration \( C \) holds significant importance. After integrating, \( C \) is added to account for any constant value that could have been differentiated to zero, ensuring that the solution is general. This addition represents the family of functions that could satisfy the integral.
When working through the steps of evaluating \( \int \frac{e^u}{(1 - e^u)^2} \, du \), remember that upon reaching the integration step itself, any constant that could have originally been part of the derivative is restored by \( C \). This ensures that the indefinite integral is complete.
The appearance of \( C \) is crucial since it embodies the indefinite nature of the integral, providing flexibility and completeness to the set of antiderivatives. Always remember to include \( C \), as it encompasses all possible primitives of the integrated function.
When working through the steps of evaluating \( \int \frac{e^u}{(1 - e^u)^2} \, du \), remember that upon reaching the integration step itself, any constant that could have originally been part of the derivative is restored by \( C \). This ensures that the indefinite integral is complete.
The appearance of \( C \) is crucial since it embodies the indefinite nature of the integral, providing flexibility and completeness to the set of antiderivatives. Always remember to include \( C \), as it encompasses all possible primitives of the integrated function.
Other exercises in this chapter
Problem 16
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle y = \int^{x^4}_0 \cos^2 \theta \,d\theta \)
View solution Problem 16
Use a calculator or computer to make a table of values of left and right Riemann sums \( L_n \) and \( R_n \) for the integral \( \displaystyle \int^2_0 e^{-x^2
View solution Problem 17
Find the general indefinite integral. \( \displaystyle \int 2^t (1 + 5^t)\, dt \)
View solution Problem 17
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle y = \int^{\pi/4}_{\sqrt{x}} \theta \tan \theta \,d\th
View solution