/*! 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 16 Find the indefinite integral and... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 16

Find the indefinite integral and check your result by differentiation. $$ \int d r $$

Short Answer

Expert verified
The indefinite integral of dr is r+C, and the result is verified to be accurate by differentiation.

Step by step solution

01

Evaluate the Indefinite Integral

The problem requires to find the indefinite integral of dr, represented as 鈭玠r. Since 'r' is a single variable, the indefinite integral of 'r' with respect to 'r' is just 'r'. So, 鈭玠r = r + C, where 'C' is the constant of integration.
02

Check the Result by Differentiation

Now, to confirm the result, differentiate the resulted function from integration, which is r+C. The derivative of 'r' with respect to 'r' is 1, and the derivative of a constant is 0. So, the derivative of (r+C) equals 1. This is the same as the original function, indicating that the result of integration is accurate.

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 Techniques
Integration is a fundamental concept in calculus that involves finding a function whose derivative is given. In the case of an indefinite integral, we are looking for a general function without limits of integration, represented by the form \( \int f(x) \, dx \). The solution to an indefinite integral is a family of functions, since integration and differentiation are inverse operations.
  • The process of solving the indefinite integral \( \int dr \) involves recognizing that the integral of a constant such as 1 (since \( dr \) can imply \( 1 \times dr \)) is \( r \), because differentiating \( r \) gives back the original function under the integral, which is 1.
  • Understanding basic integration rules can make solving integrals more intuitive. For example, when you integrate a constant \( c \), the result is \( cx \), and when you integrate \( x^n \), the result is \( \frac{x^{n+1}}{n+1} \) as long as \( n eq -1 \).
  • In the original exercise, we had a simple integral \( \int dr \), which directly results in \( r + C \). The simplicity lies in the constant 1 being integrated with respect to \( r \).
Breaking down integrals into simpler parts and understanding these basic techniques can greatly enhance your integration skillset.
Constant of Integration
When performing indefinite integration, you'll frequently see the constant \( C \) added to the solution. This constant of integration is crucial because it accounts for the fact that there are infinitely many antiderivatives for a given function.
  • Each different value of \( C \) represents a different function within the family of solutions for the integral. This means that without sometimes additional information (like boundary conditions), we cannot pinpoint a unique solution.
  • For example, in our problem \( \int dr = r + C \), \( C \) could be any real number, showing all possible vertical shifts of the function \( r \).
  • Always remember that the constant of integration is fundamental in solving integrals, as forgetting it implies a loss of generality and accuracy in solutions.
While it might seem small, the role of \( C \) is significant in creating a complete and correct representation of integrals.
Derivative Check
After solving an integral, especially indefinite ones, it's essential to verify the result through differentiation - this is known as a derivative check. By doing this, you ensure that the integration process was done correctly.
  • To perform a derivative check, take the derivative of your integrated result, and see if it equals the original function within the integral.
  • In our case, after finding \( \int dr = r + C \), we differentiate \( r + C \). The derivative of \( r \) with respect to \( r \) is 1, and the derivative of any constant \( C \) is 0. Therefore, \( \frac{d}{dr}(r + C) = 1 \), which matches the original function being integrated.
  • If the derivative matches the original function, it confirms that the integration was performed correctly.
Performing a derivative check is a simple yet powerful tool to verify your work and gives you confidence in your integration results.

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

Evaluate the definite integral. \(\int_{1}^{4} \frac{u-2}{\sqrt{u}} d u\)

Find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your results. \(y=1+\sqrt{x}, \quad y=0, \quad x=0, \quad\) and \(\quad x=4\)

Evaluate the definite integral. \(\int_{2}^{5}(-3 x+4) d x\)

Evaluate the definite integral. \(\int_{1}^{3} \frac{e^{3 / x}}{x^{2}} d x\)

Evaluate the definite integral by hand. Then use a symbolic integration utility to evaluate the definite integral. Briefly explain any differences in your results. \(\int_{2}^{3} \frac{x+1}{x^{2}+2 x-3} d x\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

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.