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Magazine Chapter 28: Problem 28
Integrate each of the given functions. $$\int \frac{4 e^{x} d x}{\left(1-e^{x}\right)^{2}}$$
Short Answer
Expert verified
The integral is \(\frac{4}{1-e^x} + C\).
Step by step solution
01
Identify the Substitution
Notice that the expression has a function of the form \(1 - e^x\). Let's set \(u = 1 - e^x\), so \(du = -e^x dx\). This substitution will simplify the integration.
02
Simplify the Integral
Rewrite the integral in terms of \(u\). From the substitution \(u = 1 - e^x\), we have \(du = -e^x dx\). Hence, \(dx = -\frac{du}{e^x}\). Actually substituting involves careful consideration: \(e^x = 1-u\).The new integral is:\[ \int \frac{4 (-du)}{u^2} \]Which simplifies to:\[ -4 \int u^{-2} \, du \]
03
Perform the Integration
Integrate \(-4 \int u^{-2} \, du\). Using the power rule for integration, we have:\[\int u^{-2} \, du = -u^{-1} + C\]Thus, the integral becomes:\[-4(-u^{-1}) = 4u^{-1} = \frac{4}{u} + C\]
04
Substitute Back the Original Variable
Recall the substitution we made: \(u = 1 - e^x\). Substitute \(u\) back in:\[\frac{4}{u} = \frac{4}{1-e^x}\]Thus, the integral becomes:\[\frac{4}{1-e^x} + C\]
05
Write the Final Answer
The integral of the given function is:\[\int \frac{4 e^x \, dx}{(1-e^x)^2} = \frac{4}{1-e^x} + C\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration by Substitution
Integration by substitution is a useful technique in calculus for simplifying integrals and finding antiderivatives. It's akin to reversing the chain rule in differentiation. The goal is to substitute part of the integral with a new variable, often denoted as \( u \), which simplifies the integration process. Here's how you apply it:
- Identify a function within the integrand that, when substituted, will simplify the integral. This is typically a component that, after differentiation, matches another part of the integrand.
- Once you have identified \( u \), compute its differential, represented as \( du \). For example, if \( u = 1 - e^x \), then differentiating gives \( du = -e^x \, dx \).
- Replace all expressions in the integral in terms of \( u \) and \( du \), translating the entire integral into a new variable.
- Perform the integration, which should be more straightforward now.
- Substitute back the original variable to express the integral in its original terms.
Power Rule for Integration
The power rule for integration is a fundamental tool for finding antiderivatives of power functions. It is the counterpart to the power rule in differentiation. The general form of the power rule for integration states:
- If \( n eq -1 \), then \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).
- This formula helps in integrating functions where the variable is raised to a constant power.
- For any integral of the form \( \int u^n \, du \), where \( n eq -1 \), apply the power rule by increasing the exponent by 1, divide by the new exponent, and add the integration constant \( C \).
Definite and Indefinite Integrals
In calculus, integrals are classified into two types: definite and indefinite. Understanding these is essential for solving a variety of problems:
- Indefinite integrals represent a family of functions and include the constant of integration \( C \). An integral without upper and lower limits, such as \( \int f(x) \, dx \), is indefinite. Its solution provides the antiderivative of the function.
- Definite integrals compute the net area under a curve, specified by the limits of integration \([a, b]\). Given by \( \int_{a}^{b} f(x) \, dx \), this type evaluates to a specific number, assuming \( f(x) \) is continuous in \([a, b]\).
