Chapter 6: Problem 21
In Exercises \(21-24,\) use tabular integration to find the antiderivative. $$\int x^{4} e^{-x} d x$$
Short Answer
Expert verified
The antiderivative of the given function is: \(x^4 e^{-x} - 4x^3 e^{-x} - 12x^2 e^{-x} + 24x e^{-x} + 24 e^{-x} + C\).
Step by step solution
01
Choose the parts for differentiation and integration
For tabular integration, one part of the function needs to be continuously differentiated until zero, while the other needs to be integrated. In this case, \(x^4\) will be continuously differentiated until it becomes zero, while \(e^{-x}\) will be integrated.
02
Carry out tabular integration
Now, carry out the tabular integration process by following the pattern of alternating signs (starting with positive). First, write down a table where the left column represents the differentiation of \(x^4\) step by step until it becomes zero, and the right column represents the integration of \(e^{-x}\) at each step.\n\n Differentiation of \(x^4\): \(x^4, 4x^3, 12x^2, 24x, 24, 0\)\n\n Integration of \(e^{-x}\): \(e^{-x}, -e^{-x}, e^{-x}, -e^{-x}, e^{-x}\)\n\n Now multiply diagonally and then sum up all the terms. Remember: alternate starting with a positive sign.\n\n Therefore, the antiderivative of \(x^{4} e^{-x}\) is \(x^4 e^{-x} - 4x^3 e^{-x} - 12x^2 e^{-x} + 24x e^{-x} + 24 e^{-x}\).\n\n Remember to include the constant of integration, \(C\), at the end.
03
Write final answer
The final antiderivative of the given function is therefore: \(x^4 e^{-x} - 4x^3 e^{-x} - 12x^2 e^{-x} + 24x e^{-x} + 24 e^{-x} + C\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Antiderivative
An antiderivative, often termed as an indefinite integral, is essentially the reverse process of differentiation in calculus. It is a function whose derivative is the original function we started with.
For instance, if you're asked to find the antiderivative of a function like \( f(x) = x^{4}e^{-x} \), you're looking for another function, let's call it \( F(x) \), such that when you differentiate \( F(x) \), you obtain \( f(x) \). When you compute the antiderivative, you should always add a constant term, denoted as \( C \), because the differentiation of a constant is zero, and it gets 'lost' during differentiation. This is why the antiderivative of a function is not unique—any constant can be added without affecting the derivative.
The task of finding antiderivatives is crucial in solving integration problems, which is a fundamental concept in areas such as physics, engineering, and economics where one might need to determine accumulated quantity from a rate of change.
For instance, if you're asked to find the antiderivative of a function like \( f(x) = x^{4}e^{-x} \), you're looking for another function, let's call it \( F(x) \), such that when you differentiate \( F(x) \), you obtain \( f(x) \). When you compute the antiderivative, you should always add a constant term, denoted as \( C \), because the differentiation of a constant is zero, and it gets 'lost' during differentiation. This is why the antiderivative of a function is not unique—any constant can be added without affecting the derivative.
The task of finding antiderivatives is crucial in solving integration problems, which is a fundamental concept in areas such as physics, engineering, and economics where one might need to determine accumulated quantity from a rate of change.
Integration by Parts
Integration by parts is a technique used in calculus to integrate products of functions. It is based on the product rule for differentiation and is formally stated by the equation \( \int u dv = uv - \int v du \), where \( u \) and \( dv \) are parts of the function chosen such that \( u \) is differentiable, and \( dv \) is integrable.
Tabular integration is a streamlined version of integration by parts that simplifies the process when applying the technique repeatedly. It is used particularly when one function in the product can be differentiated repeatedly until it reaches zero. In the given exercise, the function \( x^{4} \) becomes zero after a finite number of differentiations, making it a prime candidate for tabular integration.
An important tip for students is to choose the factors to differentiate and integrate wisely. One should typically choose a factor for differentiation that will simplify to zero after a certain number of differentiations, and a factor for integration that remains easy to integrate multiple times.
Tabular integration is a streamlined version of integration by parts that simplifies the process when applying the technique repeatedly. It is used particularly when one function in the product can be differentiated repeatedly until it reaches zero. In the given exercise, the function \( x^{4} \) becomes zero after a finite number of differentiations, making it a prime candidate for tabular integration.
An important tip for students is to choose the factors to differentiate and integrate wisely. One should typically choose a factor for differentiation that will simplify to zero after a certain number of differentiations, and a factor for integration that remains easy to integrate multiple times.
Exponential Functions in Calculus
Exponential functions, such as \( e^{-x} \) in our exercise, are a fundamental class of functions characterized by an exponent that is a variable. They are widely used across various disciplines due to their unique properties, particularly in modeling growth and decay processes.
In calculus, the differentiation and integration of exponential functions are significant because of their relative simplicity. The base of the natural exponential function, \( e \), is unique because the rate of growth of \( e^x \) is proportional to the value of \( e^x \) itself, which is reflected by the derivative of \( e^x \) being the same as the original function. When dealing with exponential functions in calculus, remember that the integral of \( e^x \) is also \( e^x \), with a constant of integration added. For a function like \( e^{-x} \), the antiderivative is \( -e^{-x} \), because the negative sign is derived from chain rule during differentiation.
In calculus, the differentiation and integration of exponential functions are significant because of their relative simplicity. The base of the natural exponential function, \( e \), is unique because the rate of growth of \( e^x \) is proportional to the value of \( e^x \) itself, which is reflected by the derivative of \( e^x \) being the same as the original function. When dealing with exponential functions in calculus, remember that the integral of \( e^x \) is also \( e^x \), with a constant of integration added. For a function like \( e^{-x} \), the antiderivative is \( -e^{-x} \), because the negative sign is derived from chain rule during differentiation.
Differentiation and Its Role
Differentiation is a cornerstone concept in calculus, representing the rate at which a quantity changes. In mathematical terms, the derivative of a function at a point is the slope of the tangent line to the function at that point.
The differentiation process is used to find the rate of change of one variable with respect to another. For example, in the given exercise, differentiating \( x^4 \) repeatedly allows us to simplify the equation through tabular integration. The function's derivatives, like \( 4x^3, 12x^2, 24x, \) and \( 24 \), provide us with the 'parts' required for tabular integration. This process showcases the interplay between differentiation and integration and emphasizes the importance of understanding both concepts to tackle calculus problems effectively.
Students should practice differentiating a variety of functions to familiarize themselves with common patterns and results, which will be advantageous when applying methods like integration by parts or tabular integration.
The differentiation process is used to find the rate of change of one variable with respect to another. For example, in the given exercise, differentiating \( x^4 \) repeatedly allows us to simplify the equation through tabular integration. The function's derivatives, like \( 4x^3, 12x^2, 24x, \) and \( 24 \), provide us with the 'parts' required for tabular integration. This process showcases the interplay between differentiation and integration and emphasizes the importance of understanding both concepts to tackle calculus problems effectively.
Students should practice differentiating a variety of functions to familiarize themselves with common patterns and results, which will be advantageous when applying methods like integration by parts or tabular integration.
