Problem 7
Question
In Problems 1-30, use integration by parts to evaluate each integral. $$ \int x e^{x} d x $$
Step-by-Step Solution
Verified Answer
The integral evaluates to \( x e^x - e^x + C \).
1Step 1: Identify Parts for Integration by Parts
Integration by parts is given by the formula: \( \int u \, dv = uv - \int v \, du \). Identify \( u \) and \( dv \) from the integral \( \int x e^x \, dx \). Here, let \( u = x \) and \( dv = e^x \, dx \).
2Step 2: Differentiate and Integrate Parts
Differentiate \( u = x \) to find \( du \) and integrate \( dv = e^x \, dx \) to find \( v \). The derivatives and integrals are: \( du = dx \) and \( v = e^x \) since the integral of \( e^x \) is itself \( e^x \).
3Step 3: Substitute into Integration by Parts Formula
Substitute \( u \), \( dv \), \( du \), and \( v \) into the formula \( \int u \, dv = uv - \int v \, du \) to get: \( \int x e^x \, dx = x e^x - \int e^x \, dx \).
4Step 4: Integrate the Remaining Integral
Calculate the integral \( \int e^x \, dx \). This integral evaluates to \( e^x \), so the expression becomes: \( x e^x - e^x + C \), where \( C \) is the constant of integration.
Key Concepts
Integral CalculusDifferentiationExponential Functions
Integral Calculus
Integral calculus is a branch of mathematics that focuses on finding the integral of functions. It helps calculate areas, volumes, and can solve problems involving accumulation. Integrals are broadly classified into definite and indefinite integrals.
In the case of indefinite integrals, such as in the problem \( \int x e^{x} dx \), we are looking for a general solution that incorporates a constant of integration \( C \). This solution allows us to represent a family of curves due to the indefinite nature of the problem.
One of the advanced techniques used in integral calculus is **integration by parts**, which is especially useful for integrating the product of two functions. Given by the formula \( \int u \, dv = uv - \int v \, du \), integration by parts breaks down a complex integral into simpler components that are easier to solve. To utilize this method, it is crucial to appropriately choose which part of the integrand will be \( u \) and which will be \( dv \).
The result of applying integration by parts often leads to another integral that is simpler or more familiar, greatly simplifying the solution process, as seen in the original exercise worksheet.
In the case of indefinite integrals, such as in the problem \( \int x e^{x} dx \), we are looking for a general solution that incorporates a constant of integration \( C \). This solution allows us to represent a family of curves due to the indefinite nature of the problem.
One of the advanced techniques used in integral calculus is **integration by parts**, which is especially useful for integrating the product of two functions. Given by the formula \( \int u \, dv = uv - \int v \, du \), integration by parts breaks down a complex integral into simpler components that are easier to solve. To utilize this method, it is crucial to appropriately choose which part of the integrand will be \( u \) and which will be \( dv \).
The result of applying integration by parts often leads to another integral that is simpler or more familiar, greatly simplifying the solution process, as seen in the original exercise worksheet.
Differentiation
Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to one of its variables. In simpler terms, it tells us how much a function changes as its input changes.
In the context of integration by parts, like in the problem \( \int x e^{x} dx \), differentiation is used to find \( du \) once we've chosen \( u \). For example, when choosing \( u = x \), the derivative \( du = dx \) represents how \( u \) changes when \( x \) is the variable.
The choice of \( u \) in integration by parts is often influenced by how easily \( du \) can be obtained, and how complex \( v \), the antiderivative of \( dv \), will become post integration. This concept is closely tied to the differentiation of functions, ensuring that each choice simplifies the problem instead of complicating it. Differentiation acts as one of the building blocks within integral calculus, providing a pathway to solving integrals that might otherwise seem daunting.
In the context of integration by parts, like in the problem \( \int x e^{x} dx \), differentiation is used to find \( du \) once we've chosen \( u \). For example, when choosing \( u = x \), the derivative \( du = dx \) represents how \( u \) changes when \( x \) is the variable.
The choice of \( u \) in integration by parts is often influenced by how easily \( du \) can be obtained, and how complex \( v \), the antiderivative of \( dv \), will become post integration. This concept is closely tied to the differentiation of functions, ensuring that each choice simplifies the problem instead of complicating it. Differentiation acts as one of the building blocks within integral calculus, providing a pathway to solving integrals that might otherwise seem daunting.
Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent, often seen in the form \( a^x \). A special and commonly used exponential function is the natural exponential function \( e^x \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828.
In the realm of calculus, exponential functions have unique properties that make them particularly significant. For instance, the derivative and the integral of \( e^x \) are both \( e^x \). This distinctive feature greatly simplifies the process of integration and differentiation, as seen in the problem where \( dv = e^{x} dx \).
The behavior of \( e^x \) doesn't change drastically through differentiation or integration, making it a preferred function to work with in many calculus problems. This stability and predictability enable mathematicians and students alike to approach problems involving \( e^x \) with confidence, knowing that the function maintains its form, enhancing both the learning and problem-solving experience.
In the realm of calculus, exponential functions have unique properties that make them particularly significant. For instance, the derivative and the integral of \( e^x \) are both \( e^x \). This distinctive feature greatly simplifies the process of integration and differentiation, as seen in the problem where \( dv = e^{x} dx \).
The behavior of \( e^x \) doesn't change drastically through differentiation or integration, making it a preferred function to work with in many calculus problems. This stability and predictability enable mathematicians and students alike to approach problems involving \( e^x \) with confidence, knowing that the function maintains its form, enhancing both the learning and problem-solving experience.
Other exercises in this chapter
Problem 7
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