/*! 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 15 Evaluate the following integrals... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 15

Evaluate the following integrals. $$\int \frac{6}{x^{2}-1} d x$$

Short Answer

Expert verified
Question: Evaluate the integral of the following rational function: $$\int \frac{6}{x^{2}-1} dx$$ Answer: $$\int \frac{6}{x^{2}-1} dx = 3\ln|x-1| - 3\ln|x+1| + C$$

Step by step solution

01

Identify the Integrals as Rational Functions

Observe that the integrand is a rational function, which is a fraction with a polynomial in the numerator and denominator. Since the degree of the polynomial in the denominator is greater than the degree of the polynomial in the numerator, we don't need to perform long division.
02

Perform Partial Fraction Decomposition

Now we need to decompose the rational function into simpler parts by using partial fraction decomposition. The denominator can be factored as \((x-1)(x+1)\). In the partial fraction decomposition, we will have: $$\frac{6}{x^2 - 1} = \frac{A}{(x-1)} + \frac{B}{(x+1)}$$ Multiplying both sides by \(x^2 - 1\), we get: $$6 = A(x+1) + B(x-1)$$
03

Solve for Constants A and B

To find the values of A and B, plug in two values for x that will eliminate one of the variables. For example, let us choose \(x = 1\) and \(x = -1\): For \(x = 1\): $$6 = A(1+1)$$ $$A = 3$$ For \(x = -1\): $$6 = B(-1-1)$$ $$B = -3$$ So we have: $$\frac{6}{x^2-1} = \frac{3}{x-1} - \frac{3}{x+1}$$
04

Integrate Term by Term

Now that the rational function is decomposed, we can integrate term by term. We have: $$\int \frac{6}{x^{2}-1} d x=\int\left(\frac{3}{x-1} - \frac{3}{x+1}\right) d x$$ The integrals can now be directly calculated as: $$\int \frac{6}{x^{2}-1} d x=3\int\frac{1}{x-1} d x -3\int\frac{1}{x+1} d x$$
05

Solve the Integrals

Apply the reverse chain rule. We have: $$\int \frac{6}{x^{2}-1} d x=3\ln|x-1| -3\ln|x+1| + C$$ Where \(C\) is the constant of integration.
06

Final Answer

The integral of the given function is: $$\int \frac{6}{x^{2}-1} d x = 3\ln|x-1| - 3\ln|x+1| + C$$

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!

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

An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution \(u=\tan (x / 2)\) or \(x=2 \tan ^{-1} u .\) The following relations are used in making this change of variables. $$A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}$$ $$\text { Evaluate } \int \frac{d x}{1-\cos x}$$

Bob and Bruce bake bagels (shaped like tori). They both make standard bagels that have an inner radius of 0.5 in and an outer radius of 2.5 in. Bob plans to increase the volume of his bagels by decreasing the inner radius by \(20 \%\) (leaving the outer radius unchanged). Bruce plans to increase the volume of his bagels by increasing the outer radius by \(20 \%\) (leaving the inner radius unchanged). Whose new bagels will have the greater volume? Does this result depend on the size of the original bagels? Explain.

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{d t}{2+e^{-t}}$$

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \sqrt{e^{x}+1} d x \text { (Hint: Let } u=\sqrt{e^{x}+1}$$

Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. $$\int \frac{d x}{x \sqrt{1+2 x}} ; 1+2 x=u^{2}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics 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.