/*! 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 14 Use partial fractions to find th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 14

Use partial fractions to find the indefinite integral. $$ \int \frac{4}{x^{2}-4} d x $$

Short Answer

Expert verified
The indefinite integral of the function is \[\ln |x-2| - \ln |x+2| + C\]

Step by step solution

01

Factoring the Denominator.

Factorize the denominator in the given integral. The factor of \(x^{2}-4\) is \((x-2)(x+2)\), since it is the difference of two squares.
02

Expressing the Fraction as a Partial Fraction.

A fraction with a quadratic term in the denominator can be expressed as the sum of two simpler fractions, or partial fractions. The actual partial fraction decomposition of \(\frac{4}{x^{2}-4}\) can be written as follows: \(\frac{4}{x^{2}-4}=\frac{A}{x-2}+\frac{B}{x+2}\),where A and B are constants.
03

Solving for the Constants A and B.

To solve for constants A and B, we make the right side identical to the left side by canceling out the denominator so that we can solve it for different values of x. We get: \(4=A(x+2)+B(x-2)\).Choosing convenient values for x, say x=2 and x=-2, will give A and B directly.Solving for A when x=2 we get:\(4=4A \Rightarrow A=1\).Next, we solve for B when x=-2:\(4=-4B \Rightarrow B=-1\).
04

Substitution and Integration.

Now substituting A and B into the partial fractions, we can integrate:\(\int \frac{4}{x^{2}-4} d x =\int \frac{1}{x-2} d x-\int \frac{1}{x+2} d x\),The integral of \(\frac{1}{x-a}\) is \(\ln |x-a|\). Hence, the result of the above integration is:\[\ln |x-2| - \ln |x+2| + C\], where C is the constant of integration.

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!

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

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{1} \frac{1}{x^{2}} d x $$

Use the error formulas to find \(n\) such that the error in the approximation of the definite integral is less than \(0.0001\) using (a) the Trapezoidal Rule and (b) Simpson's Rule. $$ \int_{1}^{3} \frac{1}{x} d x $$

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{2} \frac{1}{\sqrt{1+x^{3}}} d x, n=4 $$

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{1} \sqrt{1-x} d x, n=4 $$

Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=\frac{1}{2}, n=5 $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

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.