/*! 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 46 Solve the given problems. In Exe... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 25: Problem 46

Solve the given problems. In Exercises \(41-46\) explain your answers. Is \(\int x^{-2} d x=-\frac{1}{3} x^{-3}+C ?\)

Short Answer

Expert verified
No, the correct integral is \( -\frac{1}{x} + C \).

Step by step solution

01

Recall the Power Rule for Integration

The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). This allows us to integrate functions in the form of \( x^n \) by increasing the exponent by one and dividing by the new exponent.
02

Apply the Power Rule to Integrate

Given the integral \( \int x^{-2} \, dx \), apply the power rule: the exponent \( n = -2 \).According to the power rule, gain the new exponent by adding 1: \( n+1 = -2+1 = -1 \).Thus, the integral becomes:\[ \int x^{-2} \, dx = \frac{x^{-1}}{-1} + C = -x^{-1} + C = -\frac{1}{x} + C \]
03

Compare to Given Expression

The question asks if \( \int x^{-2} \, dx = -\frac{1}{3}x^{-3} + C \).We have calculated \( \int x^{-2} \, dx = -\frac{1}{x} + C \), which is not the same as \(-\frac{1}{3}x^{-3} + C \).The expressions are different, meaning the provided solution in the question is incorrect.

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.

Calculus Problem Solving
When tackling calculus problems, it's essential to approach each exercise with a clear understanding of the problem's requirements and a strategic plan. Calculus can often seem daunting due to its abstract nature, but following a systematic process can make problem-solving more manageable.
  • **Read the Problem Carefully**: Start by carefully interpreting what the exercise demands. Here, we aim to verify if a specific integral \(\int x^{-2} \, dx\) equals a given expression.
  • **Identify Known Formulas and Rules**: Have a clear recall of integration rules, such as the Power Rule for Integration, which we used here. Knowing these formulas inside out is crucial.
  • **Apply Logical Steps**: Use logic and the right strategy to apply these rules correctly. Formulate the solution by methodically applying known mathematical principles.
  • **Check the Solution**: After solving, compare the calculated result to the expected one to ensure accuracy. If there鈥檚 a mismatch, reevaluate the steps to find discrepancies.
Using these steps, integration problems become more approachable and less error-prone, aiding in the overall understanding of calculus.
Integration Techniques
Integration techniques are vital tools in calculus for determining the area under curves, among other applications. The Power Rule is a fundamental technique used to integrate power functions of the form \(x^n\).
  • **Understand the Power Rule**: The Power Rule states \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) when \(n eq -1\). This simple adjustment of the exponent and coefficient is fundamental for evaluating integrals involving power functions.
  • **Apply the Rule**: For example, given \(\int x^{-2} \, dx\):
    • Identify \(n = -2\).
    • Apply \(n+1 = -2+1 = -1\) to adjust the exponent.
    • Calculate the integral, resulting in \(\int x^{-2} \, dx = -\frac{1}{x} + C\).
  • **Avoid Common Mistakes**: Always remember that this rule cannot be used when \(n = -1\) due to division by zero. For \(x^{-1}\), the integral is \(\ln|x| + C\).
Mastering this rule and recognizing when and how to apply it correctly enhances your ability to tackle a broad range of integration problems effectively.
Evaluating Integrals
Evaluating integrals involves understanding both the function you are integrating and the proper application of integration techniques to achieve an accurate result.
  • **Set Up the Integral**: Carefully determine the integral form. With \(\int x^{-2} \, dx\), we are handling a basic power function suitable for the Power Rule.
  • **Perform Integration Accurately**: Methodically apply the correct rules. As in our example, using the Power Rule provided:
    Increase the exponent and divide by the new power, resulting in \(\int x^{-2} \, dx = -\frac{1}{x} + C\).
  • **Interpret the Result**: Once evaluated, ensure that the integral expression aligns with or challenges the initial hypothesis. In this exercise, comparing \(-\frac{1}{x} + C\) with the given \(-\frac{1}{3}x^{-3} + C\) helped affirm that the proposed solution was incorrect.
  • **Remain Critical**: Always cross-check your result against provided answers or expected outcomes to verify precision.
By following these steps, you can confidently and accurately evaluate integrals, enhancing your mathematical toolkit for solving complex calculus problems.

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

Approximate the value of each of the given integrals by use of the trapezoidal rule, using the given value of \(n\). $$\int_{0}^{1.5} 10^{x} d x, n=15$$

Approximate the value of each of the given integrals by use of the trapezoidal rule, using the given value of \(n\). $$\int_{0}^{4} 2^{x} d x, n=12$$

solve the given problems. In finding the average electron energy in a metal at very low temperatures, the integral \(\frac{3 N}{2 E_{F}^{3 / 2}} \int_{0}^{E_{F}} E^{3 / 2} d E\) is used. Evaluate this integral.

evaluate the given definite integrals. $$\int_{2}^{3} \frac{8\left(x^{2}+1\right)}{\left(x^{3}+3 x\right)^{2}} d x$$

In Exercises \(41-62,\) solve the given problems. The rate of change of current \(i\) (in \(\mathrm{mA}\) ) in a circuit with a variable inductance is given by \(d i / d t=300(5.0-t)^{-2},\) where \(t\) (in ms) is the time the circuit is closed. Find \(i\) as a function of \(t\) if \(i=300 \mathrm{mA}\) for \(t-2.0 \mathrm{ms}\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

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.