/*! 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 31 Use partial fractions to find th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 31

Use partial fractions to find the indefinite integral. $$ \int \frac{3 x^{2}+3 x+1}{x\left(x^{2}+2 x+1\right)} d x $$

Short Answer

Expert verified
The indefinite integral of the given function is \( - \ln|x| + \ln|x+1|- \frac{3} {x+1} + C \).

Step by step solution

01

Separate into partial fractions

Firstly, break down the expression \(\frac{3x^{2}+3x+1}{x(x^{2} + 2x + 1)}\) into its partial fractions. The denominator can be split into \(x\) and \((x + 1)^2\), thus the expression becomes \(\frac{A}{x} + \frac{B}{x+1} + \frac{C}{(x+1)^2}\). Now, set these fractions to be equal to the original expression, and solve for \(A\), \(B\), and \(C\) by comparing numerators and multiply through by the common denominator to clear it.
02

Solve for A, B, and C

We equate coefficients of the powers of \(x\) on both sides to get three equations i.e. \(A=-1\), \(B=1\) and \(C=3\). Our expression thus breaks down to \(-\frac{1}{x} + \frac{1}{x+1} + \frac{3}{(x+1)^2}\).
03

Integrate

Now, integrate each of these fractions. The integral of \(-\frac{1}{x}\) gives \(- \ln|x|\), the integral of \(\frac{1}{x+1}\) gives \(\ln|x+1|\), and the integral of \(\frac{3}{(x+1)^2}\) gives \(- 3 \/(x+1)\). Adding these together yields the indefinite integral of the original function.
04

Write final answer

Finally, add the constant of integration, denoted by \(+C\), to the result. This gives the final solution of \( - \ln|x| + \ln|x+1|- \frac{3} {x+1} + C \).

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

Use a program similar to the Simpson's Rule program on page 906 with \(n=6\) to approximate the indicated normal probability. The standard normal probability density function is \(f(x)=(1 / \sqrt{2 \pi}) e^{-x^{2} / 2}\). If \(x\) is chosen at random from a population with this density, then the probability that \(x\) lies in the interval \([a, b]\) is \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 1.5) $$

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} $$ a=\frac{1}{2}, n=5 $$

Use a program similar to the Simpson's Rule program on page 906 with \(n=6\) to approximate the indicated normal probability. The standard normal probability density function is \(f(x)=(1 / \sqrt{2 \pi}) e^{-x^{2} / 2}\). If \(x\) is chosen at random from a population with this density, then the probability that \(x\) lies in the interval \([a, b]\) is \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 4) $$

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. \int_{0}^{2} x^{2} d x, n=4

Medicine A body assimilates a 12 -hour cold tablet at a rate \(\quad\) modeled \(\quad\) by \(\quad d C / d t=8-\ln \left(t^{2}-2 t+4\right)\), \(0 \leq t \leq 12\), where \(d C / d t\) is measured in milligrams per hour and \(t\) is the time in hours. Use Simpson's Rule with \(n=8\) to estimate the total amount of the drug absorbed into the body during the 12 hours.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Pure 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.