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Chapter 6: Problem 11

Determine the following: \(\int \frac{x}{c} d x(c\) a constant \(\neq 0)\)

Short Answer

Expert verified
\(\frac{x^2}{2c} + C\)

Step by step solution

01

Understand the Integral

The given integral is \(\frac{x}{c} dx\). Recognize that \(\frac{x}{c}\) can be interpreted as \(\frac{1}{c} \times x\) because \(c\) is a constant.
02

Factor Out the Constant

Integrate \(\frac{x}{c} dx\) by first factoring out the constant \(\frac{1}{c}\), which results in \(\frac{1}{c} \times \int x \; dx\).
03

Integrate the Remaining Expression

The integral of \(x\) with respect to \x\ is given by \(\frac{x^2}{2} + C\) where \(C\) is the constant of integration. Thus, \(\frac{1}{c} \times \int x \; dx = \frac{1}{c} \times \frac{x^2}{2} + C\).
04

Simplify the Expression

Simplify the expression to obtain \(\frac{x^2}{2c} + C\).

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Key Concepts

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

definite integral
A definite integral represents the area under a curve between two specific points on the x-axis. Unlike indefinite integrals which include a constant of integration, definite integrals have limits of integration, usually denoted as \([a, b]\). The result is a number which represents the total area within those limits.
integration techniques
In calculus, integration is the reverse process of differentiation. Several techniques can be used for integration, including basic antiderivatives, substitution, integration by parts, and partial fractions. Each technique is suited for different forms of integrals. When faced with an integral, the goal is to simplify it into a form that is easier to integrate.
constant of integration
The constant of integration, denoted by \(+ C\), is a crucial part of indefinite integrals. It accounts for the fact that differentiating a function removes any constant term. Thus, when you integrate a function, you must add \(+ C\) to indicate the general solution. For definite integrals, this constant is not needed because the evaluation at the boundaries inherently accounts for any constants.
factoring out constants
Factoring out constants is a useful technique that simplifies the integration process. When you have a constant multiplied by a function inside an integral, you can move the constant outside of the integral sign. This simplifies the problem and makes it easier to focus on integrating the remaining function on its own. For example, in \(\frac{1}{c} \times \frac{x^2}{2} + C\), the constant \(c\) was factored out initially before integrating.

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Most popular questions from this chapter

A property with an appraised value of $$\$ 200,000$$ in 2008 is depreciating at the rate \(R(t)=-8 e^{-.04 t}\), where \(t\) is in years since 2008 and \(R(t)\) is in thousands of dollars per year. Estimate the loss in value of the property between 2008 and 2014 (as \(t\) varies from 0 to 6 ).

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