/*! 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 23 Find the indefinite integral and... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 23

Find the indefinite integral and check the result by differentiation. $$ \int \frac{x}{\sqrt{1-x^{2}}} d x $$

Short Answer

Expert verified
\(\frac{3}{5} x^{5/3} + C \)

Step by step solution

01

Integration

The integral is given by \( \int x^{2/3} dx \). We can integrate this function by using the power rule for integration, which states that \( \int x^n dx = \frac{x^{n+1}}{n+1}\) for any number n that's not -1. In this case, n = 2/3. So, applying the power rule gives \( \frac{x^{5/3}}{(5/3)} + C = \frac{3}{5} x^{5/3} + C \). 'C' is the constant of integration, which arises because the integral of any constant is a constant.
02

Differentiation

To prove that the obtained solution is correct, differentiate the result. We use the power rule for differentiation, which is \( \frac{d(x^n)}{dx} = n x^{n - 1}\). Apply this rule to \( \frac{3}{5} x^{5/3} \) and remember that 'C' goes away because the derivative of a constant is zero. Doing this will give the original integrand \( x^{2/3} \), confirming that we correctly integrated the function.

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.

Power Rule for Integration
The power rule for integration is a fundamental rule that helps us find antiderivatives or indefinite integrals. It's especially useful for algebraic functions where the exponent is not equal to -1. When applied, this rule converts the process of integration into a simple formula to follow. Here's how it works:
  • For any function of the form \( x^n \), where \( n eq -1 \), the integral is given by \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).
  • The expression \( C \) is added as the constant of integration, which we'll explore more in the next section.
In the original exercise, \( n \) was \( 2/3 \), so using the power rule, we calculated the indefinite integral of \( \sqrt[3]{x^2} \) to be \( \frac{3}{5} x^{5/3} + C \). This simple formula allows us to integrate power functions without much hassle. Remember, the key is ensuring that \( n \) is anything but -1, as that scenario would require a different approach.
Constant of Integration
Whenever we calculate an indefinite integral, we must add a constant of integration, denoted as \( C \). This constant is crucial because integration is essentially the reverse process of differentiation, and a constant term vanishes when differentiating. Here are some key points to understand:
  • The constant \( C \) represents any arbitrary constant value, as there are infinitely many functions that could differentiate into the same result.
  • It accounts for all possible vertical shifts of a curve represented by the integrated function.
  • In problems requiring a specific solution, initial conditions or boundaries are needed to determine \( C \).
In the context of our exercise, the expression \( \frac{3}{5} x^{5/3} + C \) reflects not just one function, but a family of functions that differ by a constant value. Because differentiating any constant results in zero, we include \( C \) to maintain the integrity of indefinite integration solutions.
Power Rule for Differentiation
The power rule for differentiation is a straightforward rule that helps find the derivative of functions where the variable is raised to a power. Consider the function \( x^n \):
  • The derivative of \( x^n \) is given by \( \frac{d(x^n)}{dx} = n \cdot x^{n-1} \).
  • This rule simplifies finding the rate of change for any polynomial or power function quickly.
To verify the integration result in the original exercise, we differentiated \( \frac{3}{5} x^{5/3} + C \). Using the power rule,
  • We multiplied the power \( 5/3 \) by the coefficient \( 3/5 \), simplifying to 1.
  • The power of \( x \) decreased by 1, becoming \( 2/3 \).
  • Additionally, since the derivative of \( C \) is zero, it was simply omitted.
Thus, the differentiation confirmed that our integral of \( x^{2/3} \) led to the correct original integrand, confirming the accuracy of the integration using these foundational rules.

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

Find the indefinite integral and check the result by differentiation. $$\int(2 \sin x+3 \cos x) d x$$

Show that the function $$ f(x)=\int_{0}^{1 / x} \frac{1}{t^{2}+1} d t+\int_{0}^{x} \frac{1}{t^{2}+1} d t $$ is constant for \(x>0\).

The maker of an automobile advertises that it takes 13 seconds to accelerate from 25 kilometers per hour to 80 kilometers per hour. Assuming constant acceleration, compute the following. (a) The acceleration in meters per second per second (b) The distance the car travels during the 13 seconds

Complete the table. Original Integral $$\int x\left(x^{2}+3\right) d x$$

Solve the differential equation. $$f^{\prime \prime}(x)=x^{2}, f^{\prime}(0)=6, f(0)=3$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Decision Maths

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.