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Magazine Chapter 6: Problem 32
Find each integral by using the integral table on the inside back cover. $$ \int \frac{e^{2 t}}{1-e^{t}} d t $$
Short Answer
Expert verified
The integral is \(- \ln |1 - e^t| + 2(1 - e^t) - \frac{(1 - e^t)^2}{2} + C.\)
Step by step solution
01
Identifying the Integral Type
The given integral is \( \int \frac{e^{2t}}{1-e^t} \, dt \). By examining the form, we notice that it resembles the form \( \int \frac{e^{at}}{b - e^{ct}} \, dt \), which often requires substitution to simplify.
02
Substitution
To simplify, use the substitution \( u = 1 - e^t \), which implies that \( du = -e^t \, dt \). Therefore, \( e^t \, dt = -du \) and \( e^{2t} = (1-u)^2 \). Substituting these into the integral gives us: \[ \int \frac{(1-u)^2}{u} (-du) = -\int \frac{(1-u)^2}{u} \, du. \]
03
Simplifying the Integral
Distribute \((1-u)^2\) over \(u\) to get \[ - \int \left( \frac{1}{u} - 2 + u \right) \, du. \] This breaks into three separate integrals: \[ - \int \frac{1}{u} \, du + 2 \int du - \int u \, du. \]
04
Evaluating the Integrals
Evaluate each of the integrals separately: - \( - \int \frac{1}{u} \, du = - \ln |u| \),- \( 2 \int du = 2u \), and- \( - \int u \, du = - \frac{u^2}{2}. \)
05
Combining Results
Combine the results from the integrals to obtain:\[ - \ln |u| + 2u - \frac{u^2}{2} + C. \]
06
Back Substitution
Replace \(u\) with the original variable: \( u = 1 - e^t \). Thus, the integral becomes: \[ - \ln |1 - e^t| + 2(1 - e^t) - \frac{(1 - e^t)^2}{2} + C. \] Simplify if possible.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integral Table
Integral tables are incredibly handy when evaluating or solving integrals. They provide a collection of standard integral forms and their corresponding results. This is particularly useful for integrals that can have complicated or non-standard forms.
The integral table acts similar to a dictionary for integrals. Instead of solving complex integrals from scratch, you can look up certain standard forms and use the solutions provided.
The integral table acts similar to a dictionary for integrals. Instead of solving complex integrals from scratch, you can look up certain standard forms and use the solutions provided.
- Identify the form of your integral by examining its structure.
- Locate the corresponding form in the integral table.
- Apply any listed solutions as a guide to find your integral's solution.
Substitution Method
The substitution method, often referred to as "u-substitution," is a technique used to simplify integrals into a more solvable form. It is akin to changing variables in order to make the integral more manageable.
- First, identify which part of the integral can be replaced with a new variable \( u \).
- Differentiate \( u \) to find \( du \) and replace the corresponding part of the integral.
- Carry out the integral with respect to the new variable \( u \).
Definite and Indefinite Integrals
Understanding the difference between definite and indefinite integrals is crucial in calculus. Each serves a different purpose and result in distinct solutions.
- An indefinite integral represents a family of functions and includes a constant \( C \). Writing it as \( \int f(x) \, dx = F(x) + C \) signifies the general antiderivative of the function.
- A definite integral calculates the net area under a curve between two specific points. It is written as \( \int_{a}^{b} f(x) \, dx \) and results in a numerical value, representing the accumulation of the function's values over the interval \( [a, b] \).
