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Chapter 6: Problem 18

Find the general antiderivative. $$h(x)=x^{3}-x$$

Short Answer

Expert verified
The general antiderivative is \( \frac{x^4}{4} - \frac{x^2}{2} + C \).

Step by step solution

01

Identify the function to be integrated

The function given is \( h(x) = x^3 - x \). We need to find its antiderivative, which is the function whose derivative is \( h(x) \).
02

Integrate each term separately

The function can be separated into its components: \( x^3 \) and \( -x \). We integrate each component term by term: \[ \int (x^3) \, dx = \frac{x^4}{4} + C_1 \]\[ \int (-x) \, dx = -\frac{x^2}{2} + C_2 \]
03

Combine the antiderivatives

Add the two antiderivatives together to form the general antiderivative:\[ \int h(x) \, dx = \frac{x^4}{4} - \frac{x^2}{2} + C \]where \( C = C_1 + C_2 \) is the constant of integration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration
Integration is a fundamental concept in calculus that is often viewed as the reverse process of differentiation. While differentiation is about finding the rate of change or the slope of a curve, integration is all about finding the area under a curve. This is particularly useful in a wide variety of applications such as physics, engineering, and economics.

Here are some key points about integration:
  • Indefinite vs. Definite Integrals: An indefinite integral is the antiderivative of a function and includes a constant of integration (denoted as 'C'). A definite integral, on the other hand, is calculated over a specific interval and gives a numerical value.
  • The Process: To integrate a function like the one in the original exercise, the function is typically broken down into simpler components that are easier to work with.
  • Symbol: The integral symbol \( \int \) is used to denote the operation of integration, and 'dx' indicates that we are integrating with respect to the variable x.
Understanding these points provides a strong foundation in approaching problems that involve integration.
Calculus
Calculus is a branch of mathematics that focuses on the study of change. It consists of two major ideas: differentiation and integration. While differentiation looks at how functions change, integration considers the accumulation of quantities.

Calculus is used to understand and create models that require an understanding of:
  • Derivatives: Which measure the rate of change of a quantity.
  • Integrals: Which accumulate or combine quantities to find totals.
  • Limits: A concept integral to both differentiation and integration, providing the basis for defining both derivatives and integrals.
Through its applications, calculus enables the solving of complex problems and answering questions about real-world changes and accumulations. This makes it a powerful tool in fields ranging from sciences to finance.
Indefinite Integral
An indefinite integral, often known as an antiderivative, is a type of integration where we find a family of functions that can have many solutions. The general form of an indefinite integral is represented by \( \int f(x) \, dx = F(x) + C \), where \( F(x) \) is the antiderivative of \( f(x) \), and \( C \) is the constant of integration.

Some important aspects of indefinite integrals include:
  • Constant of Integration: The constant 'C' is crucial because differentiation of a constant is zero, which implies that many functions can share the same derivative.
  • Finding Antiderivatives: The process can usually be done term by term for polynomials, trigonometric, and exponential functions using standard rules of integration.
  • General Solutions: Indefinite integrals are used to find general solutions of functions, which are essential in solving problems where specific initial conditions are unknown.
Understanding how to compute indefinite integrals is foundational in solving calculus problems that involve reversing the process of differentiation.

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Most popular questions from this chapter

Evaluate the definite integrals exactly \([as in \)\ln (3 \pi)]\(,\) using the Fundamental Theorem, and numerically \([\ln (3 \pi) \approx 2.243]\) $$\int_{0}^{2} 3 e^{x} d x$$

Are the statements true or false? Give an explanation for your answer. If \(F(x)=\int_{0}^{x} f(t) d t\) and \(G(x)=\int_{0}^{x} g(t) d t,\) then \(F(x)+G(x)=\int_{0}^{x}(f(t)+g(t)) d t\).

Give an example of: Two different solutions to the differential equation $$\frac{d y}{d t}=t+3$$

Give an example of: A function \(G(x),\) constructed using the Second Fundamental Theorem of Calculus such that \(G\) is concave up and \(G(7)=0\).

(a) Explain why you can rewrite \(x^{x}\) as \(x^{x}=e^{x \ln x}\) for \(x>0\) (b) Use your answer to part (a) to find \(\frac{d}{d x}\left(x^{x}\right)\) $$\text { (c) Find } \int x^{x}(1+\ln x) d x$$ (d) Find \(\int_{1}^{2} x^{x}(1+\ln x) d x\) exactly using part (c). Check your answer numerically.
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