/*! 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 Find each indefinite integral. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 31

Find each indefinite integral. \(\int \frac{z^{3}+z}{z^{2}} d z\)

Short Answer

Expert verified
The indefinite integral is \(\frac{z^2}{2} + \ln|z| + C\).

Step by step solution

01

Simplify the Expression

First, simplify the integrand. The given integral is \(\int \frac{z^3 + z}{z^2} \, dz\).Divide each term in the numerator by \(z^2\):\(\frac{z^3}{z^2} = z\) and \(\frac{z}{z^2} = \frac{1}{z}\).So, the integral becomes \(\int (z + \frac{1}{z}) \; dz\).
02

Integrate Each Term Separately

Now integrate each term separately:\(\int z \, dz\) is a basic power rule integration problem, which gives \(\frac{z^2}{2}\).\(\int \frac{1}{z} \, dz\) is a standard integral, giving \(\ln|z|\).
03

Combine the Results and Add the Constant

Combine the results of the integrations from Step 2:The integral \(\int (z + \frac{1}{z}) \, dz = \frac{z^2}{2} + \ln|z| + C\), 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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration Techniques
When solving indefinite integrals, choosing the right integration technique is crucial. It helps you find the antiderivative of a function. For rational functions like \( \int \frac{z^3+z}{z^2} \, dz \), there are specific techniques that simplify the process:
  • **Simplify First:** Before integrating, always look to simplify the integrand. This might involve factorization or simplifying fractions, as in this case.
  • **Integrate Term by Term:** Once simplified, integrate each term of the expanded expression separately if possible.
Using the right technique ensures accurate solutions and helps develop a deeper understanding of calculus methods.
Simplification of Algebraic Expressions
Simplifying algebraic expressions is a key step in calculus. It makes complex integrals more manageable. Consider the expression \( \frac{z^3 + z}{z^2} \). Here, simplification involves dividing each term in the numerator by \( z^2 \).
  • Divide \( z^3 \) by \( z^2 \) to yield \( z \).
  • Divide \( z \) by \( z^2 \) to yield \( \frac{1}{z} \).
This results in the expression \( z + \frac{1}{z} \), which is much easier to integrate. Simplification turns the problem into two simple terms. Successfully simplifying expressions often depends on recognizing common algebraic structures and applying rules for division. This practice not only simplifies calculations but also leads to deeper insights into the behavior of functions.
Integral of Rational Functions
Integrating rational functions can seem daunting, but breaking them into simpler parts helps. Unlike other functions, rational functions often have polynomial numerators and denominators. In our integral \( \int (z + \frac{1}{z}) \, dz \), each term is straightforward:
  • \( \int z \, dz \) uses the power rule, resulting in \( \frac{z^2}{2} \).
  • \( \int \frac{1}{z} \, dz \) leads to the natural logarithm, \( \ln|z| \).
Rational functions often involve this combination of techniques. Once each term is integrated, results are combined, plus a constant of integration \( C \). Knowing these techniques allows for efficient solutions to problems involving rational integrals, boosting both skill and confidence in technical problem-solving.

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

BEHAVIORAL SCIENCE: Repeated Tasks A subject can perform a task at the rate of \(\sqrt{2 t+1}\) tasks per minute at time \(t\) minutes. Find the total number of tasks performed from time \(t=0\) to time \(t=12\).

85-94. The substitution method can be used to find integrals that do not fit our formulas. For example, observe how we find the following integral using the substitution \(u=x+4\) which implies that \(x=u-4\) and so \(d x=d u\). $$ \begin{aligned} \int(x-2)(x+4)^{8} d x &=\int(u-4-2) u^{8} d u \\ &=\int(u-6) u^{8} d u \\ &=\int\left(u^{9}-6 u^{8}\right) d u \\ &=\frac{1}{10} u^{10}-\frac{2}{3} u^{9}+C \\ &=\frac{1}{10}(x+4)^{10}-\frac{2}{3}(x+4)^{9}+C \end{aligned} $$ It is often best to choose \(u\) to be the quantity that is raised to a power. The following integrals may be found in this way (as well as by the methods of Section 6.1). $$ \int(x+1)(x-5)^{4} d x $$

\mathbf{4 5}-\mathbf{6 0} \text { . Find the area bounded by the given curves. } y=x^{2}-1 \text { and } y=2-2 x^{2}

A friend says that finding differentials is as easy as finding derivatives-you just multiply the derivative by \(d x\). Is your friend right?

70-74 \text { . Find the derivative of each function. } \(e^{x^{3}+6 x}\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.