/*! 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 19 In Exercises \(5-28\) , find the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 19

In Exercises \(5-28\) , find the indefinite integral. $$\int \frac{x^{3}-3 x^{2}+5}{x-3} d x$$

Short Answer

Expert verified
The indefinite integral of the given function is \((1/3)x^{3}+3x^{2}-5x+22\ln|x-3|+C\).

Step by step solution

01

Polynomial Division

Divide the polynomial \(x^{3}-3 x^{2}+5\) by \(x-3\). This step simplifies the function before integration. It provides us with a form that is more amenable to integration.
02

Write out the division result as a new function for integration

The result of the division in step 1 will yield a new simpler function. Express the original integral as an integral of this new simpler function. The new representation is \(\int (x^{2}+6x-5+22/(x-3)) d x\).
03

Separate into Separate Integrals

Separate the new function into separate simpler function which can be easily integrated. This is done based on the fact that the integral of a sum (or difference) of functions is equal to the sum (or difference) of their integrals. Therefore the new expression becomes \(\int (x^{2})d x+\int (6x) dx -\int (5) dx +\int(22/(x-3)) d x\).
04

Find the Indefinite Integration

Perform the integration operation for each separate function obtained in step 3. The integral of \(x^{2}\) is \((1/3)x^{3}\), the integral of \(6x\) is \(3x^{2}\), the integral of \(5\) is \(5x\), and the integral of \(22/(x-3)\) is \(22\ln|x-3|\). Add the constant of integration, \(C\), at the end.

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.

Polynomial Division
Just like dividing numbers, polynomial division breaks down a complex polynomial into simpler parts. We use it when a polynomial in our integral's numerator is more complex than the denominator.
In this exercise, we have \(x^3 - 3x^2 + 5\) divided by \(x - 3\). By dividing, we simplify the polynomial expression, making the integration process smoother later on.
This division also helps by separating terms into a polynomial part and a remainder, which is essential for integration, especially when dealing with rational functions.
Integration Techniques
Integration involves finding a function whose derivative matches the given function. For rational functions like the one in this exercise, separation into simpler terms helps a lot.
Here, after polynomial division, the expression \((x^2 + 6x - 5 + \frac{22}{x-3})\)\ allows us to separate it into parts that are easier to integrate independently.
  • The process highlights the fact that integrating sums or differences of functions can often be done term by term.
  • You can directly integrate each part using basic integration rules.
  • It simplifies integration significantly, avoiding complex substitutions or special functions.
Integral of a Rational Function
Rational functions are ratios of polynomials and often appear in calculus problems. When attempting to find the integral of a rational function, separating the expression into simpler integrable parts is key.
From our example, we divide and separate \(\int \frac{x^3 - 3x^2 + 5}{x-3} dx\) into distinct integrals like \(\int x^2 dx\), \(\int 6x dx\), and so on, resulting from the polynomial division.
  • The term \(\int \frac{22}{x-3} dx\) specifically requires recognizing it as a logarithmic function, which comes from inverse relationships in calculus.
  • This often involves remembering that integrating 1/x forms \(-ln|x|\), modified by coefficients.
  • Each part ties back to fundamental integration techniques, reinforcing their utility in problems involving rational functions.

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

Analyzing a Graph Consider the function \(f(x)=\frac{2}{1+e^{1 / x}}\) (a) Use a graphing utility to graph \(f .\) (b) Write a short paragraph explaining why the graph has a horizontal asymptote at \(y=1\) and why the function has a nonremovable discontinuity at \(x=0\) .

In Exercises 47-50, find the indefinite integrals, if possible, using the formulas and techniques you have studied so far in the text. $$\begin{array}{l}{\text { (a) } \int \frac{1}{1+x^{4}} d x} \\ {\text { (b) } \int \frac{x}{1+x^{4}} d x} \\ {\text { (c) } \int \frac{x^{3}}{1+x^{4}} d x}\end{array}$$

Inverse Secant Function Some calculus textbooks define the inverse secant function using the range \([0, \pi / 2) \cup[\pi, 3 \pi / 2).\) (a) Sketch the graph \(y=\operatorname{arcsec} x\) using this range. (b) Show that \(y^{\prime}=\frac{1}{x \sqrt{x^{2}-1}}\)

Trigonometric Functions Integrating which trigonometric function results in \(\ln \ln |\sin x|+C ?\)

In Exercises 43-46, use the specified substitution to find or evaluate the integral. $$\begin{array}{l}{\int_{1}^{3} \frac{d x}{\sqrt{x}(1+x)}} \\\ {u=\sqrt{x}}\end{array}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

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.