/*! 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 32 Determine the following indefini... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 32

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(4 z^{1 / 3}-z^{-1 / 3}\right) d z$$

Short Answer

Expert verified
Question: Find the indefinite integral of the function $$\int\left(4 z^{1 / 3}-z^{-1 / 3}\right) d z$$. Answer: The indefinite integral of the given function is $$3z^{4/3} - \frac{3}{2}z^{2/3} + C$$.

Step by step solution

01

Identify the terms to be integrated separately

Given the function $$\int\left(4 z^{1 / 3}-z^{-1 / 3}\right) d z$$, we can separate the two terms and integrate them separately. They are: 1. $$4z^\frac{1}{3}$$ 2. $$-z^{-\frac{1}{3}}$$
02

Integrate the first term

We will integrate $$4z^{1/3}$$ with respect to z. Use the power rule of integration, which states that $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$, where n is a constant. $$\int 4z^{1/3} dz = 4\int z^{1/3} dz = 4\cdot\frac{z^{1/3+1}}{1/3+1}+C_1$$ Simplify the expression: $$= 4\cdot\frac{z^{4/3}}{4/3}+C_1$$ $$= 3z^{4/3} + C_1$$
03

Integrate the second term

Next, we will integrate $$-z^{-1/3}$$ with respect to z. Again, use the power rule of integration: $$\int -z^{-1/3} dz = -\int z^{-1/3} dz = -\frac{z^{-1/3+1}}{-1/3+1}+C_2$$ Simplify the expression: $$= -\frac{z^{2/3}}{2/3}+C_2$$ $$= -\frac{3}{2}z^{2/3} + C_2$$
04

Combine the results

Now, we will combine the results from Step 2 and Step 3 to find the indefinite integral: $$\int(4 z^{1 / 3}-z^{-1 / 3}) d z = 3z^{4/3} - \frac{3}{2}z^{2/3} + C$$ where $$C = C_1 + C_2$$ is the constant of integration.
05

Verify the answer by differentiation

To check our work, we will differentiate the result and see if it matches the original function. $$\frac{d}{dz} (3z^{4/3} - \frac{3}{2}z^{2/3} + C)$$ Using the power rule for differentiation, we get: $$\frac{d}{dz} (3z^{4/3}) - \frac{d}{dz} (\frac{3}{2}z^{2/3})$$ $$= 4z^{1/3} - z^{-1/3}$$ The differentiated result matches the original function, so our integration is correct.

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

Locate the critical points of the following functions and use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. $$f(x)=x^{3}+2 x^{2}+4 x-1$$

Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. $$f(x)=2 x^{3}-3 x^{2}+12$$

Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. $$f(x)=x^{2} e^{-x}$$

Graph carefully Graph the function \(f(x)=60 x^{5}-901 x^{3}+27 x\) in the window \([-4,4] \times[-10000,10000] .\) How many extreme values do you see? Locate all the extreme values by analyzing \(f^{\prime}\)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If \(f^{\prime}(x)>0\) and \(f^{\prime \prime}(x)<0\) on an interval, then \(f\) is increasing at a decreasing rate on the interval. b. If \(f^{\prime}(c)>0\) and \(f^{\prime \prime}(c)=0,\) then \(f\) has a local maximum at \(c\) c. Two functions that differ by an additive constant both increase and decrease on the same intervals. d. If \(f\) and \(g\) increase on an interval, then the product \(f g\) also increases on that interval. e. There exists a function \(f\) that is continuous on \((-\infty, \infty)\) with exactly three critical points, all of which correspond to local maxima.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

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.