/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 298 For the following exercises, fin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 298

For the following exercises, find the indefinite integral. $$\int \frac{d t}{3 t}$$

Short Answer

Expert verified
\( \int \frac{d t}{3 t} = \frac{1}{3} \ln |t| + C.\)

Step by step solution

01

Recognize the Integrand Format

The integrand can be recognized as a fraction where the numerator is simply the differential \(dt\) and the denominator is \(3t\). This suggests the form \(\frac{1}{3} \int \frac{1}{t} \, dt\).
02

Pull Out Constant Factor

Observe that \(\frac{1}{3}\) is a constant factor. We can bring this constant outside the integral as follows: \(\frac{1}{3} \int \frac{1}{t} \, dt\).
03

Integrate Using Basic Logarithm Rule

Recall the integral rule for \(\int \frac{1}{x} \, dx = \ln |x| + C\). Apply this rule to integrate:\[\frac{1}{3} \int \frac{1}{t} \, dt = \frac{1}{3} \ln |t| + C,\]where \(C\) is the constant of integration.
04

Final Step: Present the Solution

Combine the results to present the final solution:\[\int \frac{d t}{3 t} = \frac{1}{3} \ln |t| + C.\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Integration by Substitution
Integration by substitution is a method used when the integrand is a composite function. This technique typically simplifies the integral by making a substitution to transform the integrand into an easier form. To apply substitution:
  • Identify a part of the integrand as a new variable, usually denoted as "\(u\)".
  • Substitute this new variable into the integrand, replacing the original variable.
  • Once the integral is in terms of \(u\), integrate with respect to \(u\).
  • After integration, substitute back the original variable to express the final result.
This is particularly helpful for integrals that involve products or compositions of functions, where one function is the derivative of another. However, in the original problem \(\int \frac{d t}{3 t}\), integration by substitution isn't necessary, as the integrand is already in a straightforward form for logarithmic integration.
Logarithmic Integration
Logarithmic integration is applicable when encountering an integrand of the form \(\frac{1}{x}\). This is because there is a well-known rule for integrating this form:
  • The integral of \(\frac{1}{x}\) is \(\ln |x| + C\), where \(C\) is the constant of integration.
In the original exercise, the integrand is \(\frac{1}{t}\) multiplied by a constant factor, \(\frac{1}{3}\). This can be rewritten as \(\frac{1}{3} \int \frac{1}{t} \, dt\). The integration rule is straightforward, allowing us to apply the logarithmic rule directly:
  • Integrate \(\frac{1}{t}\) to get \(\ln |t|\),
  • Then multiply by the constant \(\frac{1}{3}\).
Therefore, the integral of \(\frac{d t}{3 t}\) is \(\frac{1}{3} \ln |t| + C\).
Constant Factor in Integration
A constant factor in an integral can be placed outside the integral sign, making the process simpler. This technique relies on a property of definite and indefinite integrals, allowing algebraic constants to be factored out:
  • If you have an integral of the form \(\int k \cdot f(x) \, dx\), it can be rewritten as \(k \cdot \int f(x) \, dx\), where \(k\) is a constant.
In the exercise, the given integrand is \(\frac{1}{3} \cdot \frac{1}{t}\). The constant factor \(\frac{1}{3}\) was taken out of the integral, which simplifies the integral into \(\frac{1}{3} \cdot \int \frac{1}{t} \, dt\). This factorization simplifies the integration process, and only the core function \(\frac{1}{t}\) needs to be integrated, while the constant remains untouched outside multiplying the resultant integral.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For the following exercises, consider the catenoid, the only solid of revolution that has a minimal surface, or zero mean curvature. A catenoid in nature can be found when stretching soap between two rings. Find surface area of the catenoid \(y=\cosh (x)\) from \(x=-1\) to \(x=1\) that is created by rotating this curve around the \(x\) -axis.

True or False? If true, prove it. If false, find the true answer. You are cooling a turkey that was taken out of the oven with an internal temperature of \(165^{\circ} \mathrm{F}\) . After 10 minutes of resting the turkey in a \(70^{\circ} \mathrm{F}\) apartment, the temperature has reached \(155^{\circ} \mathrm{F}\) . What is the temperature of the turkey 20 minutes after taking it out of the oven?

True or False? If true, prove it. If false, find the true answer. The effect of advertising decays exponentially. If 40\(\%\) of the population remembers a new product after 3 days, how long will 20\(\%\) remember it?

True or False? If true, prove it. If false, find the true answer. The populations of New York and Los Angeles are growing at 1\(\%\) and 1.4\(\%\) a year, respectively. Starting from 8 million (New York) and 6 million (Los Angeles), when are the populations equal?

For the following exercises, find a. the area of the region, b. the volume of the solid when rotated around the \(x-\) axis, and c. the volume of the solid when rotated around the \(y\) -axis. Use whichever method seems most appropriate to you. \(y=x^{2}\) and \(y=\sqrt{x}\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.