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Chapter 6: Problem 12

Determine the following: $$\int x \cdot x^{2} d x$$

Short Answer

Expert verified
\int x^3 \, dx = \frac{x^4}{4} + C

Step by step solution

01

- Simplify the Integrand

Combine the powers of x in the integrand. The integrand can be rewritten as \( x \times x^2 = x^{1+2} = x^3 \). Therefore, the integral becomes \[ \int x^3 \, dx \].
02

- Apply the Power Rule

Use the power rule for integration, which states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \, where n \) is a constant. For \( x^3 \, n = 3 \). Thus, we get: \int x^3 \, dx = \frac{x^{3+1}}{3+1} + C = \frac{x^4}{4} + C \.

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Key Concepts

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

integral
Integration is a fundamental concept in calculus that involves finding the area under a curve. It is often represented with the integral symbol (鈭�). The integral of a function can represent various real-world concepts, such as the total distance traveled given a velocity function. In our exercise, we are integrating a polynomial function.
power rule
The power rule is a simple yet powerful tool for integrating functions of the form \( x^n \). This rule states: \[ \ \ \int x^n dx = \frac{x^{n+1}}{n+1} + C \ \ \ \] where \( n \) is a constant, and \( C \) is the constant of integration. In our example, applying the power rule to \( x^3 \) involves increasing the exponent by one and then dividing by the new exponent.
simplifying integrand
Before applying the power rule, it is crucial to simplify the integrand. Simplifying the integrand makes the integration process straightforward and error-free. In the original problem, the integrand is \( x \times x^2 \). By combining the powers of \( x \), we have: \[ x \times x^2 = x^{1+2} = x^3 \ \ \] This simplifies the integral to a form where the power rule can be directly applied.
constants of integration
The constant of integration, denoted by \( C \), appears whenever we integrate. It represents an unknown constant added to the function because the process of differentiation (the reverse of integration) cancels out constants. Without the constant of integration, the solution would be incomplete, as it would only represent one particular antiderivative among infinitely many possible ones. Therefore, our final answer, integrating \( x^3 \), includes \( C \), leading to: \[ \frac{x^4}{4} + C \ \ \]

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Most popular questions from this chapter

Evaluate a Riemann sum to approximate the area under the graph of \(f(x)\) on the given interval, with points selected as specified. \(f(x)=\sqrt{1-x^{2}} ;-1 \leq x \leq 1, n=20,\) left endpoints of subintervals.

A company's marginal cost function is \(.1 x^{2}-x+12\) dollars, where \(x\) denotes the number of units produced in 1 day. (a) Determine the increase in cost if the production level is raised from \(x=1\) to \(x=3\) units. (b) If \(C(1)=15,\) determine \(C(3)\) using your answer in (a).

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