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Chapter 6: Problem 8

Determine the following: $$\int \frac{x}{3} d x$$

Short Answer

Expert verified
\(\int \frac{x}{3} \, dx = \frac{x^2}{6} + C\)

Step by step solution

01

Identify the integral to be solved

The integral given is \(\int \frac{x}{3} \, dx\).
02

Factor out constant coefficient

Factor out the constant \(\frac{1}{3}\) from the integrand. This gives us: \[\frac{1}{3} \int x \, dx.\]
03

Integrate the function

Integrate \(x \) with respect to \(x\). The integral of \(x\) is \[\int x \, dx = \frac{x^2}{2}.\]
04

Combine the results

Multiply the result by the constant factor \(\frac{1}{3}\): \[\frac{1}{3} \cdot \frac{x^2}{2} = \frac{x^2}{6}.\]
05

Add the constant of integration

Don't forget to add the constant of integration \(C\): \[\int \frac{x}{3} \, dx = \frac{x^2}{6} + C.\]

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Key Concepts

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

indefinite integrals
Indefinite integrals are a fundamental concept in integral calculus. When we calculate an indefinite integral, we look for a function whose derivative matches the original function. This is also known as finding the antiderivative.
For example, if we are given a function like \(\frac{x}{3}\), we need to determine a function whose differentiation would yield the original function.
Indefinite integrals do not have specified bounds, and they include a constant of integration, denoted as \(+ C\). This constant represents all the possible vertical shifts of the antiderivative graph.
constant coefficient
A constant coefficient in a function is a number that multiplies a variable. When integrating a function with a constant coefficient, we can factor it out of the integral to simplify the process.
  • In our example, \(\frac{x}{3}\) has a constant coefficient of \(\frac{1}{3}\).
  • We rewrite the integral as \[ \frac{1}{3} \int x \, dx \].
  • This step makes it easier to integrate the remaining part of the function.
Factoring out the constant coefficient is a helpful strategy to streamline the integration process.
integration by parts
Integration by parts is a specific technique used when integrating the product of two functions. It relies on the formula:
\[ \int u \, dv = uv - \int v \, du \].
  • In this method, you select parts of the integrand (the function you are integrating) to assign to \(u\) and \(dv\).
  • The goal is to simplify the integral into a more manageable form.
However, in the given exercise, we do not need to use integration by parts because we deal with a direct, simple integral.
antiderivative
An antiderivative is a function that reverses differentiation. It is another way of describing an indefinite integral.
  • Finding the antiderivative is essential for resolving integrals.
  • In our example, the antiderivative of \(x\) is \(\frac{x^2}{2}\).
  • After factoring out the constant coefficient and integrating, we multiply the result by the constant coefficient again, and add the constant of integration \(+ C\).
This process ensures that we have correctly found the indefinite integral with all necessary components included.

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Most popular questions from this chapter

Find the value of \(k\) that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How?] $$\int 3 e^{t / 10} d t=k e^{t / 10}+C$$

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