Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 17: Problem 1

By expressing the following in partial fractions evaluate the given integral. Remember to select the correct form for the partial fractions. $$ \int \frac{1}{x^{3}+x} \mathrm{~d} x $$

Short Answer

Expert verified
Question: Evaluate the integral \(\int \frac{1}{x^{3}+x} \mathrm{~d} x\). Answer: The integral evaluates to \(\int \frac{1}{x^{3}+x} \mathrm{~d} x = \ln|x| + C\).

Step by step solution

01

Factor the Denominator

To factor the denominator, we can first factor out an x: $$ x^3 + x = x(x^2 + 1) $$
02

Set Up the Partial Fractions

We express the given function as a sum of partial fractions: $$ \frac{1}{x(x^2 + 1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} $$
03

Find A, B, and C

To find the coefficients A, B, and C, we multiply both sides by the common denominator \(x(x^2 + 1)\): $$ 1 = A(x^2 + 1) + (Bx + C)x $$ Now, let's solve for the coefficients: - Let x=0: $$ 1 = A\cdot(0+1) \Rightarrow A=1 $$ - Compare x^2 coefficients: $$ 0 = B \Rightarrow B=0 $$ - Compare constant coefficients: $$ 1 = A+C \Rightarrow C=0 $$ Thus, the partial fractions decomposition is: $$ \frac{1}{x(x^2 + 1)} = \frac{1}{x} + \frac{0x + 0}{x^2 + 1} = \frac{1}{x} $$
04

Integrate the Partial Fractions

Now we can integrate the partial fractions: $$ \int \frac{1}{x^{3}+x} \mathrm{~d} x = \int \frac{1}{x} \mathrm{~d} x $$ The integration of \(\frac{1}{x}\) is straightforward: $$ \int \frac{1}{x} \mathrm{~d} x = \ln|x| + C $$ So, the solution to the given integral is: $$ \int \frac{1}{x^{3}+x} \mathrm{~d} x = \ln|x| + C $$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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 (a) \(\int x \sin (2 x) \mathrm{d} x\), (b) \(\int t \mathrm{e}^{3 t} \mathrm{~d} t\) (c) \(\int x \cos x \mathrm{~d} x\).

Find \(\int_{0}^{\pi / 2} \cos ^{2} t \mathrm{~d} t\).

Find \(\int \frac{\sin ^{3} x}{\cos x}+\sin x \cos x \mathrm{~d} x\).

Find \(\int 3 \cos n \pi x \mathrm{~d} x\).

By expressing the following in partial fractions evaluate the given integral. Remember to select the correct form for the partial fractions. $$ \int \frac{1}{(x+1)(x-5)} \mathrm{d} x $$
See all solutions

Recommended explanations on Physics Textbooks

Quantum Physics

Read Explanation

Fluids

Read Explanation

Mechanics and Materials

Read Explanation

Circular Motion and Gravitation

Read Explanation

Engineering Physics

Read Explanation

Conservation of Energy and Momentum

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.

Sign-up for free